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TOPIC – 6 PROBABILITY

THEORYProbability means chance of happening of an Event.The chance of happening of an Event is expressed as a number from 0 to 1.When the event is a sure event its probability is 1When the event is impossible its probability = 0

Important terms.1.Experiment.

An Experiment means a situation or an activity which has more than one outcome.Examples of experiments are Tossing a coinThrowing a dicePlaying a cricket matchAppearing in an examination.An experiment is said to be random if the result or outcome of the experiment cannot be predicted.

2.Sample space. Sample space means a list of all possible outcomes of an experiment.The number of items listed in sample space is denoted by n(s)

Experiment Outcomes n(s)Tossing a coin (H) (T) 2Tossing two coins (HH) (HT)(TH)(TT) 4Throwing three coins (HHH) (HHT) (HTH)(HTT)(THH)(THT)(TTH)(TTT) 8Throwing a Dice 1, 2, 3, 4, 5, 6 6Throwing two dice (1,1) (1,2) (1,3)------------(1,6)

(2,1) (2,6)

(6,6) 36

3. Event. Event means a particular outcome or a Set of outcomes from the sample space. Events are Denoted by Symbols A, B C etc.The number of outcomes covered by the event is denoted by n(A) , n(B) etc.Event is a sub set of Sample space.N(A) <= n(S)

4. Probability of an event = n(A)/n(S)Q1 Write short Notes on Following types of events.1 Mutually Exclusive events2. Collective exhaustive events: 3) Complementary events:4) Independent events and dependents events:

1) Mutually exclusive events Two events are said to be mutually exclusive events or disjoint when both cannot happen simultaneously at the same time. In other words occurrence of one excludes the possibility of occurrence of another. For e.g If a bag contains Red, White and black balls and if one ball is drawn. In such a case 3 events are possible, either we may draw a Red ball or a white ball or a black ball. But if a Red ball is drawn, we cannot get the white or black ball and so on. Such events are called mutually exclusive events. Drawing a Red ball and white ball and black ball are mutually exclusively event. The probability of simultaneous happening to 2 mutually exclusive events is O i.e. p (A) n (B) = 0.

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2) Collective exhaustive events : Events are said to be collective exhaustive when the events are such that all the possible outcomes are covered by the events. For e.g tossing a dice, let the first event be getting a odd number and the second event be getting a even number. Both the event cover all the possible outcomes because outcome 1,3,5 are covered by 1st event and 2,4,6 are covered by second event.If 2 event are collective exhaustive then P(A) U p(B) = 1 i.e the event collectively exhausts all the possible outcomes.If two events are collectively exhaustive P(A) U P(B) = 1If two events A and B are mutually exclusive and collectively exhaustive in such a case P(A) + P(B)=1

3) Complementary events :If 2 events are such that they are mutually exclusive as well as collective exhaustive. For e.g. when a coin is tossed, and the events considered are getting a head or getting a tail. Both the events are mutually exclusive because both the events cannot happen together. Such events are collective exhaustive because, these two events cover all the possible outcomes.

4) Independent events and dependents events :Two or more events are said to be independent when the outcome of one is not affected by other. For e.g. if a coin is tossed up ten times the result of each toss is not affected by previous results. However, some times events are such that the happening or non happening of an event depends upon the happening or non happening of other event. Let A be 1st event be firm spending a large amount of money on advertisement and the 2nd event be increase in sales. The second is likely to be take place if the 1st event has taken place.Suppose two cards is drawn from a pack of cards.Let A = Card is Diamond B = Card is SpadeThe probability of B is 13/52. However if the first card drawn is Diamond in such a case the probability that the second card is spade is 13/51. There for happening of A changes the Probability of happening of B If the first card is drawn and then replace in such a case the probability of B is 13/52.

If for two events A and B, are (a) Independent Then P (A B) P(A) X P (B), then the two events A and B

5) Equally likely events. Two events A and B are equally likely if P(A) = P(B).

Q2. State the addition therom and multiplication therom of Probability.ADDITION THEOROM.The addition theorem states that if the two events A and B are mutually exclusive the probability of the occurrence of either A or B is the sum of the individual probability of A and BIn other wordsP( A OR B) = P(A) + P(B)the probability that the card is a club or diamond.Let a be the event that the card is a club card p(a) = 13/52Let b be the event that the card is a diamond card p(b) = 13/52The events are mutually exclusive i.e. if the card is a club card then it cannot be a diamond card.

P( A or B) = 13/52 + 13/52 = 26/52 = 0.5The above equation is not applicable when the events are not mutually exclusive. When the event are not mutually exclusive there is a possibility of the occurrence of both the events at the same time.

In such a case P( A OR B) = P(A) + P(B) - P ( A n B)For example if a card is drawn from a pack of cards we would like to find the probability that the card is a CLUB or QUEEN.

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Let A be the event that the card is a club card P(A) = 13/52Let B be the event that the card is a QUEEN card P(B) = 4/52The events are NOT mutually exclusive i.e. Both the events can happen together. The Probability of both the events happening together is

P(A n B) = 1/52 P (A or B) = 13/52 + 4/52 – 1/52 = 16/52.

Multiplication theorem. According to this theorem if two events A and B are independent then the probability of happening of both the events is the product of their individual probabilities.

P( A and B) = P(A) x P(B) For example the probability that a girl is fair is 1/20the probability that the girl is rich is 1/500The girl is educated is 1/10In such a case if Mr A wants to marry a girl who is fair rich and educated the probability of finding such a girl is 1/20 x 1/500 x 1/10 ie 1/1,00,000

If A and B are independent events than Probability of A n B = P(A) X P(B)If A and B are dependent events than probability of A n B = P(A) XP(B/A)

Q3 Write short note on conditional probability.The conditional probability of an event means the probability of occurrence of an event when it is known that other related event has already occurred. Conditional probability is important in case of dependent event. Suppose A and B are dependent events such that if A occurs and the chance of occurrence of B increases in such a case the probability of occurrence of B is the conditional probability.Let A be the event that a person suffers from MalariaLet B be the event that a person dies.Now the Probability that a person will die is P(B). Such a probability is unconditional probability as it is the probability independent of other events.

However the Probability that a person will die given that he is suffering from malaria will be higher such an probability is chance of occurrence of an event under certain circumstance this probability is called as conditional probabilityThe conditional probability of B given that A has already occurred is denoted by P(B\A).

P(B/A) = P(A n B)

------------ P(A)

P(A/B) = P(A n B)------------ P(B)

If P(A/B) = P(A) then Events A and B are independentAlso if P(B/A) =P(B) then the Events A and B are independent.

Q4. Distinguish between Simple events and compound eventsSimple event means happening of a single event. Compound event means happening of two or more simple events together. For example if A is the event that it will rain today is a simple event.If B is an event that the day is Sunday and it will rain is a compound event.

Q5. What is an composite event.An event which can be split in to more than one events is called as composite Event.

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Q6. Explain the meaning of the term “Odds in favour of “and “Odds against”Odds in favour of means ratio of Happening of an event to the Non Happening of an event. For example odds in favour of A winning a Match is 5: 3 it means A will win 5 times and Awill not win three times.

Therefore P(A) 5/8P(A’) – 3/8

Odds against means ratio of Non Happening of an event to Happening of an event.

For example odds against A solving a problem is 4: 1.It means A will not solve 4 times but will solve 1 timeP(A solving )= 1/5P(A not solving) =4/5

Set A Module theoretical questions

1. Initially, probability was a branch of(a) Physics (b) Statistics (c) Mathematics (d) Economics

2. Two broad divisions of probability are(a) Subjective probability and objective probability(b) Deductive probability and non-deductive probability(c) Statistical probability and Mathematical probability(d) None of these

3. Subjective probability may be used in(a) Mathematics (b) Statistics (c) Management (d) Accountancy

4. An experiment is known to be random if the results of the experiment.(a) Can not be predicted (b) Can be predicted(c) Can be split into further experiments (d) Can be selected at random

5. An event that can be split into further events is known as(a) Complex event (b) Mixed event(c) Simple event (d) Composite event

6. Which of the following pairs of events are mutually exclusive?(a) A : The student reads in a school B : He studies Philosophy(b) A : Raju was born in India B : He is a fine Engineer(c) A : Ruma is 16 years old B : She a good singer(d) A : Peter is under 15 years of age B : Peter is a voter of Kolkata

7. If P (A) = P (B), then(a) A and B are the same events (b) A and B must be same events(c) A and B may be different events(d) A and B are mutually exclusive events

8. If P (A B) = 0, then the two event A and B are(a) Mutually exclusive (b) Exhaustive(c) Equally likely (d) Independent

9. If for two events A and B, P (AUB) = 1, then A and B are(a) Mutually exclusive events (b) Equally likely events(c) Exhaustive events (d) Dependent events

10. If an unbiased coin is tossed once, then the two events Head and Tail are(a) Mutually exclusive (b) Exhaustive(c) Equally likely (d) All these (a), (b) and (c)

11. If P (A) = P (B), then the two events A and B are(a) Independent (b) Dependent(c) Equally likely (d) Both (a) and (c)

12. If for two events A and B, P (A B) P(A) X P (B), then the two events A and B are(a) Independent (b) Dependent

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(c) Not equally likely (d) Not exhaustive

13. If P (A/B) = P (A), then(a) A is independent of B (b) B is independent of A(c) B is dependent of A (d) Both (a) and (b)

14. If two events A and B are independent, then(a) A and the complement of B are independent(b) B and the complement of A are independent(c) Complements of A and B are independent(d) All of these (a), (b) and (c).

15. If two events A and B are independent, then(a) They can be mutually exclusive(b) They can not be mutually exclusive(c) They can not be exhaustive(d) Both (b) and (c)

16. If two events A and B are mutually exclusive, then(a) They are always independent (b) They may be independent(c) They can not be independent (d) They can not be equally likely

17. If a coin is tossed twice, then the events ‘occurrence of one head’, ‘occurrence of 2 heads’ and ‘occurrence of no head’ are(a) Independent (b) Equally likely(c) Not equally likely (d) Both (a) and (b)

18. The probability of an event can assume any value between(a) -1 and 1 (b) 0 and 1 (c) -1 and 0 (d) none of these

19. If P(A) = 0, then the event A(a) will never happen (b) will always happen(c) may happen (d) may not happen

20. If P(A) = 1, then the event A is known as(a) symmetric event (b) dependent event(c) improbable event (d) sure event

21. If p:q are the odds in favour of an event, then the probability of that event is

(a) p/q (b) (c) (d) none of these

22. I f P (A) = 5/9, then the odds against the event A is(a) 5 : 9 (b) 5 : 4 (c) 4: 5 (d) 5 : 14

23. If A, B and C are mutually exclusive and exhaustive events, then P(A) + P(B) + P (C) equals to(a) 1/3 (b) 1 (c) 0 (d) any value between 0 and 1

24. If A denotes that a student is reading. and B denotes that student is sleeping, then(a) P (A B) = 1 (b) P (A B) = 1(c) P (A B) = 0 (d) P (A) = P (B)

25. P (B /A) is defined only when(a) A is a sure event (b) B is a sure event(c) A is not an impossible event (d) B is an impossible event

26. P (A/B’) is defined only when(a) B is not a sure event (b) B is a sure event(c) B is an impossible event (d) B is not an impossible event

27. For two events A and B, P (A B) = P (A) + P(A) only when(a) A and B are equally likely events (b) A and B are exhaustive events(c) A and B are mutually independent (d) A and B are mutually exclusive

28. Addition Theorem of probability states that for any two events A and B,(a) P (A B) = P (A) + P(B)(b) P (A B) = P (A) + P (B) + P (A B)(c) P (A B) = P (A) + P (B) – P (A B)

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(d) P (A B) = P (A) x P (B)

29. For any two events A and B(a) P (A) + P (B) > P (A B) (b) P (A) + P (B) < P (A B)(c) P (A) + P(B) P (A B) (d) P (A) x P (B) P (A B)

30. For any two events A and B,(a) P (A –B) = P (A) – P (B)(b) P (A – B) = P (A) – P (A B)(c) P (A – B) = P (B) – P (A B)(d) P (B – A) = P (B) + P (A B)

31. The limitations of the classical definition of probability(a) it is applicable when the total number of elementary events is finite(b) it is applicable if the elementary events are equally likely(c) it is applicable if the elementary events are mutually independent(d) (a) and (b)

32. According to the statistical definition of probability, the probability of an event A is the (a) limiting value of the ratio of the no. of times the event A occurs to the

number of times the experiment is repeated.(b) the ratio of the frequency of the occurrence of A to the total frequency(c) the ratio of the frequency of the occurrence of A to the non-occurrence

of A(d) the ratio of the favourable elementary events to A to the total number of

elementary events.

33. The Theorem of Compound Probability states that for any two events A and B.(a) P (A B) = P (A) x P (B/A) (b) P (A B) = P (A) x P (B/A)(c) P (A B) = P (A) x P (B) (d) P (A B) = P (B) + P (B) – P (A B).

34. If A and B are mutually exclusive events, then(a) P (A) = P (A – B) (b) P (B) = P (A – B)(c) P (A) = P (A B) (d) P (B) = P (A B)

35. If P (A – B) = P (B – A), then the two events A and B satisfy the condition(a) P (A) = P(B) (b) P (A) + P (B) = 1(c) P (A B) = 0 (d) P (A B) = 1

36. The number of conditions to be satisfied by three events A, B and C for independence is (a) 2 (b) 3 (c) 4 (d) any number

37. If two events A and B are independent, then P (A B)(a) equals to P (A) + P (B) (b) equal to P (A) x P (B )(c) equals to P (A) x P (B/A) (d) equals to P (B) x P (A / B)

38. Values of a random variable are(a) always positive numbers (b) always positive real numbers(c) real numbers (d) natural numbers

39. Expected value of a random variable(a) is always positive (b) may be positive or negative(c) may be positive or negative or zero (d) can never be zero

40. If all the values taken by a random variable are equal then(a) its expected value is zero(b) its standard deviation is zero(c) its standard deviation is positive (d) its standard deviation is a real number

41. If x and y are independent, then(a) E (xy) = E(x) x E (y) (b) E (xy) = E(x) + E(y)(c) E (x+y) = E(x) + E (y) (d) E(x-y) = E(x) – x E (y)

42. If a random variable x assumes the values x1, x2, x3, x4 with corresponding probabilities P1, P2, P3, P4 then the expected value of x is(a) P1+ P2+ P3+ P4 (b) X1 P1+ X2 P3+ X3 P2+ X4 P4

(c) P1X1 + P2 X2 + P3 X3 + P4 X4 (d) none of these

43. f (x), the probability mass function of a random variable x satisfies

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(a) f (x) > 0 (b) f (x) = 1

(c) both (a) and (b) (d) f(x) 0 and 1

44. Variance of a random variable x is given by(a) E (x - )2 (b) E [X – E (x)]2

(c) E (x2 - ) (d) (a) or (b)

45. If two random variables x and y are related by y = 2 – 3x, then the SD of y is given by(a) -3 x SD of x (b) 3 X SD of x(c) 9 x SD of x (d) 2 X SD of x

46. Probability of getting a head when two unbiased coins are tossed simultaneously is(a) 0.25 (b) 0.50 (c) 0.20 (d) 0.75

47. If an unbiased coin is tossed twice, the probability of obtaining at least one tail is(a) 0.25 (b) 0.50 (c) 0.75 (d) 1.00

48. If an unbiased die is rolled once, the odds in favour of getting a point which is a multiple of 3 is(a) 1 : 2 (b) 2 : 1 (c) 1 : 3 (d) 3 : 1

49. A bag contains 15 one rupee coins, 25 two rupee coins and 10 five rupee coins. If a coin is selected at random from the bag, then the probability of not selecting a one rupee coin is(a) 0.30 (b) 0.70 (c) 0.25 (d) 0.20

50. A, B, C are three mutually independent with probabilities 0.3, 0.2 and 0.4 respectively. What is P (A B C)?(a) 0.400 (b) 0.240 (c) 0.024 (d) 0.500

51. If two letters are taken at random from the word HOME, what is the Probability that none of the letters would be vowels?(a) 1/6 (b) 1/2 (c) 1/3 (d) 1/4

52. If a card is drawn at random from a pack of 52 cards, what is the chance of getting a Spade or an ace?(a) 4/13 (b) 5/13 (c) 0.25 (d) 0.20

53. If x and y are random variables having expected values as 4.5 and 2.5 respectively, then the expected value of (x-y) is(a) 2 (b) 7 (c) 6 (d) 0

54. If variance of a random variable x is 23, then what is the variance of 2x + 10?(a) 56 (b) 33 (c) 46 (d) 92

55. What is the probability of having at least one ‘six’ from 3 throws of a perfect die?(a) 5/6 (b) (5/6)3 (c) 1-(1/6)3 (d) 1 – (5/6)3

Unit I - Problems on Simple Events

Q1. Write the sample space for the event that 3 coins are tossed.

Q2. Two fair coins are tossed at a time Write the sample space of this random experiment and state the number of points in the sample space.

Q3. A perfect cubic die is thrown. Find the probability that an even number comes up.

Q4. A perfect cubic die is thrown. Find the probability that a perfect square comes up.

Q5. If n(S) = 36, P(A) = 0.5 Find n(A)

Q6. A card is drawn from a well shuffled pack of 52 playing cards.

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Find the probability that is not a face card

Q7. A card is drawn from a well shuffled pack. Find the probability of getting a face card.

Q8. Two unbiased dice are thrown. Find the probability that the sum of the scores on their upper faces is 8.

Q9. A fair die is thrown. Find the probability of getting a number on the upper face which is (i) even (ii) greater than 2.

Q10. A bowl contains 100 slips numbered 1 to 100. A slip is drawn at random from the bowl. Find the probability that the slip bears a number which is divisible by 5.

Q11. A bowl contains 100 slips numbered 1 to 100. A slip is drawn at random from the bowl. Find the probability that the slip bears a number which is divisible by 7.

Q12. Two dices are thrown what is the probability that the sum of the scores on the uppermost faces is i) Seven ii) Six

Q13. Given below are the weekly wages (in Rupees) of six workers in a factory: 62 90 78 85 79 68

If two of these workers are selected at random to serve as representatives, what is the probability that at least one will have a wage lower than the average?

Unit 1 : Basic Sums

Q1. There are 10 balls numbered from 1 to 10 in a box. If one of them is selected at random, what is the probability that the number printed on the ball would be an odd number greater that 4?(a) 0.50 (b) 0.40 (c) 0.60 (d) 0.30

Q2. What is the probability that a leap year selected at random would contain 53 Saturdays?(a) 1/7 (b) 2/7 (c) 1/12 (d) 1/4

Q3. Following are the wages of 8 workers in rupees:50, 62 , 40, 70, 45, 56, 32, 45If one of the workers is selected at random, what is the probability that his wage would be lower than the average wage?(a) 0.625 (b) 0.500 (c) 0.375 (d) 0.450

Q4. If an unbiased coin is tossed three, what is the probability of getting more that one head?(a) 1/8 (b) 3/8 (c) 1/2 (d) 1/3

Q5. A number is selected at random from the first 1000 natural numbers. What is the probability that the number so selected would be a multiple of 7 or 11?(a) 0.25 (b) 0.32 (c) 0.22 (d) 0.33

PROBLEMS ON ADDITION THEOREM

Q6. From a well shuffled pack of 52 playing cards, a card is drawn at random. Find the probability that the card drawn is a diamond or a face card.

Q7. A card is drawn at random from a pack of 52 cards. Find the probability that the card is (i) an ace or a king (ii) an ace or a spade.[(i)2/13 (ii) 4/13]

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Q8. A card is drawn at random from a pack of 52 cards. Find the probability that the card is an ace or a heart. (4/13)

Q9. There are 100 lottery tickets numbered form 1 to 100. A ticket is drawn at random. Find the probability that the numbers on the card is divisible by 5 or 7.

Q10. Two fair dice are thrown. Find the probability that the sum of the scores on the upper faces is at least 11.

Q11. Two fair dice are thrown. Find the probability that the sum of the scores is divisible by 2 or 5.

Q12. Two fair dice are thrown. Find the probability that the sum of the scores is an even number or a perfect square.

Q13. There are 15 tickets bearing numbers from 1 to 15 in a bag. One ticket is drawn from the bag at random. Find the probability that is the ticket bears a number which is even or a multiple of 4.

Q14. A card is drawn at random from a pack of 52 cards. Find the probability that the card is red or bears a number between 5 and 10 both exclusive.

Q15. One lottery ticket is drawn at random from a bag containing 20 tickets, numbered from 1 to 20. Find the probability that the number on the ticket drawn is (i) either even or square of an integer (ii) divisible by 3 or 5 (iii) odd number or multiple of 3. [(i)3/5 (ii)9/20 (iii)13/20]

Unit 2 Addition theorem

Q16 That is the chance of throwing at least 7 in a single cast with 2 dice?(a) 5/12 (b) 7/12 (c) 1/4 (d) 17/36

Q17. A, B and C are three mutually exclusive and exhaustive events such that

P (A) = 2 P (B) = 3P(C). What is P (B)?(a) 6/11 (b) 6/22 (c) 1/6 (d) 1/3

Q18. A bag contains 12 balls which are numbered from 1 to 12. If a ball is selected at random, what is the probability that the number of the ball will be a multiple of 5 or 6?(a) 0.30 (b) 0.25 (c) 0.20 (d) 1/3

Q19. If two unbiased dice are rolled, what is the probability of getting points neither 6 nor 9?(a) 0.25 (b) 0.50 (c) 0.75 (d) 0.80

Unit 3 Problems on Multiplication theoroem

Q20. One lottery ticket is drawn at random from a set of 25 tickets numbered 1 to 25. Find the probability that the number on the ticket drawn is (i) either odd or square of an integer(ii) multiple of 4 or 5 (iii) even number or divisible by 5

[(i)3/5 (ii)2/5 (iii)3/5]

Q21. The probability that a man will be alive for 60 years is 3/5 and that his wife will be alive for 60 years is 2/3 Find the probability that,i) Both will be alive for 60 years.ii) Only the man will be alive for 60 years.iii) None will be alive for 60 years.

Q22. Two cards are drawn at random from a pack of 52 playing cards. Find the number of points in the sample space for this random experiment.

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Q23. Two cards are drawn from pack of 52 well shuffled cards. Find the probability that FIRST is a king and the SECOND is a queen.

Q24. Two cards are drawn from a pack of well shuffled cards find the probability that one is a spade and the other is an ace.

Q25. From a pack of 52 cards, two cards are drawn at random. Find the probability that both are spade cards.

Q26. Two Cards are drawn from a pack of 52 playing cards. One by one without replacement. Find the probability that the

I) first card drawn is a king and second is not a king.II) One is king and other is queen.

Q27. A bag contains, 7 blue balls and 10 yellow balls, 2 balls are drawn at random. Find the probability that both are of different colour.

Q28. The probability that A can solve a problem is 1/2 and that B can solve the same problem is 2/3, if both of them try independently, find the probability that the problem is solved.

Q29. A bag contains 6 white and 9 black balls. If three balls are drawn at random find the probability that all are black.

Q30. Two cards are drawn at random from a pack of 52 cards. Find the probability that they are of different suits.

Q31. The chance of A winning a race is 1/6 and chance of B winning a race is 1/8. What is the chance that neither of the two should win?

Q32. Two cards are drawn one after the other from a pack of 52 cards. Find the probability that both the cards are kings, when(i) The first card is replaced.(ii) The first card is not replaced.

Q33. The probability that a student ‘A’ can solve a problem is 1/3 ‘B’ can solve it is 1/2 and ‘C’ can solve it is 1/4, if all of them try independently, what is the probability that the problem is solved.

Q34. A problem is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. Find the probability that the problem will not be solved considering that they are trying independently.

Q35. A purse contains 4 silver coins and 5 copper coins. Another purse contains 3 silver and 4 copper coins. A purse is selected at random and a coin is drawn from it at random. What is the probability that it is a copper coin?

Q36. An urn contains 3 white and 5 red balls and another urn contains 2 white and 4 red balls. One urn is selected at random and a ball is drawn from it at random. Find the probability that the ball is red.

Q37. An urn contains 3 red and 4 black balls and another urn contains 2 red and 4 black balls. One urn is selected at random and a ball is drawn from it at random. Find the probability that the ball is red.

Q38. Three children Seeta, Geeta and Raju often help their parents in the kitchen. The respective probability of their breaking a dish in the kitchen are 1/3 , 1/2 , and 4/5, if one of them is selected at random to help with the dishes. Find the probability that a dish broken.

Q39. Two students appear for an examination. Their chances of passing the examination being 0.7 and 0.8 respectively. Find the probability that atleast one of them passes the examination.

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Q40. A box contains 10 radio valves of which 4 are defective. Find the probability that if two valves are taken from the box, they are both defective.

Q41. A bag contain 6 blue,4 white and 5 purple marbles. Three marbles are taken out at random, What is the probability that i)all the three are blue. ii)the three are of three different colours.

Q42. A box contains 5 blue,6 black and 8 green marbles are drawn at random what is the probability that i)two are blue and 1 black ii)two are green and 1 black iii)all the three are black. iv) One of each colour.

Q43. A housewife buys a dozen eggs of which two are bad. She chooses 4 eggs to scramble for breakfast. Find the probability that she chooses i)All good eggs; ii)three good and 1 bad egg; iii)2 good and 2 bad eggs; iv)at least 1 bad egg.

Q44. From a group of 5 men and 4 women, 4 persons are selected at random to form a committee. What is the probability that the committee contains 3 men and a woman?

Q45. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. Find the probability that three cards contain two kings and one ace.

Q46. A room has 3 electric lamps. From a selection of 12 electric bulbs of which 8 are good, 3 bulbs are selected at random and put in the sockets. Find the room is lighted by at least one of bulbs.

Q47. An urn contains 5 blue and an unknown number x of red balls. When two balls are drawn at random the probability of both of them being blue is 5/14 find x.

Unit 3 Multiplication theorem

Q48. Two balls are drawn from a bag containing 5 white and 7 black balls at random. What is the probability that they would be of different colours?(a) 35/66 (b) 30/66 (c) 12/66 (d) None of these

Q49. What is the chance of getting at least one defective item if 3 items are drawn randomly from a lot containing 6 items of which 2 are defective item?(a) 0.30 (b) 0.20 (c) 0.80 (d) 0.50

Q50. If A, B and C are mutually exclusive independent and exhaustive events then that is the probability that they occur simultaneously.(a) 1 (b) 0.50 (c) 0 (d) any value between 0 and 1

Q51. If two unbiased dice are rolled together, what is the probability of getting no difference of points?(a) 1/2 (b) 1/3 (c) 1/5 (d) 1/6

Q52. It is given that a family of 2 children has a girl, what is the probability that the other child is also a girl?(a) 0.50 (b) 0.75 (c) 1/3 (d) 2/3

Q53. Tow coins are tossed simultaneously. What is the probability that the second coin would show a tail given that the first coin has shown a head?(a) 0.50 (b) 0.25 (c) 0.75 (d) 0.125

Q54. What is the probability that 4 children selected at random would have different birthdays?

(a) (b)

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(c) 1/365 (d) (1/7)3

Q55. A box contains 5 white and 7 blacks balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively.(a) 6/321 and 3/926 (b) 1/20 and 1/30

(c) 35/144 and 35/108 (d) 7/968 and 5/264

Q56. There are three boxes with the following composition:Box I: 5 Red + 7 White + 6 Blue ballsBox II: 4 Red + 8 White + 6 Blue ballsBox III: 3 Red + 4 White + 2 Blue ballsIf one ball is drawn at random, then that is the probability that they would be of same colour?(a) 89/727 (b) 97/729 (c) 82/729 (d) 23/32

Q57. A bag contains 8 red and 5 white balls. Two successive draws of 3 balls are made without replacement. The probability that the first draw will produce 3 white balls and the second 3 red balls is(a) 5/223 (b) 6/257 (c) 7/429 (d) 3/548

Q58. There are two boxes containing 5 white and 6 blue balls and 3 white and 7 blue balls respectively. If one of the boxes is selected at random and a ball is drawn from it, then the probability that the ball is blue is(a) 115/227 (b) 83/250 (c) 137/220 (d) 127/250

Q59. A problem in probability was given to three CA students A, B and C whose chance of solving it are 1/3, 1/5 and 1/2 respectively. What is the probability that the problem would be solved?(a) 4/15 (b) 7/8 (c) 8/15 (d) 11/15

Q60. There are three persons aged 60,65 and 70 years old. The survivals probabilities for these three persons for another 5 years are 0.7, 0.4 and 0.2 respectively. What is the probability that at least two of them would survive another five years?(a) 0.425 (b) 0.456 (c) 0.392 (d) 0.388

Q61. Tom speaks truth in 30 percent cases and Dick speaks truth in 25 percent cases. What is the probability that they would contradict each other?(a) 0.325 (b) 0.400 (c) 0.925 (d) 0.075

Q62. There are two urns. The first urn contains 3 red and 5 white balls whereas the second urn contains 4 red and 6 white balls. A ball is taken at random from the first urn and is transferred to the second urn. Now another ball is selected at random from the second arm. The probability that the second ball would be red is(a) 7/20 (b) 35/88 (c) 17/52 (d) 3/20

Q63. A packet of 10 electronic components is known to include 2 defectives. If a sample of 4 components is selected at random from the packet, what is the probability that the sample does not contain more than 1 defective?(a) 1/3 (b) 2/3 (c) 13/15 (d) 3/15

Q64. 8 identical balls are placed at random in three bags. What is the probability that the first bag will contain 3 balls?(a) 0.2731 (b) 0.3256 (c) 0.1924 (d) 0.3443

Q65. X and Y stand in a line with 6 other people. What is the probability that there are 3 persons between them?(a) 1/5 (b) 1/6 (c) 1/7 (d) 1/3

UNIT 4 - PROBLEMS ON SET THEORY

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Q66. Given: P(A B)= 5/9 P(A B)= 1/36 P(A) = 12/36 Find P(B)

Q67. Given:- P(A UB)= 1/2 , n(S) = 8, Find n(A U B)

Q68. Given:- P(A B) = 0.25, n(A B) = 13, Find n(S).

Q69. If A and B are two events in a sample space S such that P(A) = 0.8, P(B) = 0.6 and P(A n B) = 0.5 Find P(A U B) and P(A/B)

Q70. There are 100 students in a class. 50 pass in Mathematics,40 in physics and 10 in both. If a student is selected at random what is the probability B that he has passed in i)at least one subject ii)in one subject only, iii)in both the subjects iv)in one of the subjects v)only in physics.

Q71. The probability that a person stopping at a petrol pump will ask for petrol is 0.80 and the probability that he will ask for water is 0.70 and the probability that he will ask for both is 0.65. Find the probability that a person stopping at this petrol pump will ask for neither petrol nor water.

Q72. There are 80 members of a gymkhana. 60 of them play table tennis,30 play lawn tennis and 15 play both the games. If a member is selected at random, what is the probability that he plays i)table tennis ii)only table tennis iii) both the games iv) neither of the two games.

Q73. 30 per cent of the families in a city have T.V. and 40 per cent have telephone. If 25 per cent of those who have telephone have T.V. also, what is the probability that a family selected at random from that city will not have T.V. but have telephone?

Q74. The department of a company has records which show the following analysis of its 200 MembersAge Bachelor’s Master’s Total

Degree DegreeUnder 30 90 10 100 30 to 40 20 30 50over 40 40 10 50Total 150 50 200If one engineer is selected at random from the company find

a) The probability he has only a bachelor’s degreeb) The Probability he has a master’s Degree given that he is over 40 c) The probability he is under 30 given that he has only a bachelor’s degree.

Q75. Calculate Pr. (B\A) if (A) = 0.75 Pr.(B) = 0.60 and Pr (A\B)=0.90 Q76. A and B are two events such that P(A)= 0.8, P(B) = 0.6,

P(A B)=0.5. Find (i)P(A U B) (ii) P(A/B) (iii) P(B/A)

Q77. P(A) = 0.3, P(B)=0.4 and P(A/B)= 0.32. Find (i)P(A B) (ii)P(B/A)

Q78. A and B are two events such that P(A) = 2/3, P((B’) = 3/4, P(A/B)= 4/5, Find P(A B) and P(B/A)

Q79. It is known that 20 % of the males and 5% of the females are unemployed in a certain town consisting of an equal number of males and females. A person is selected at random and is found to be unemployed What is the probability that he is i) a male ii) a female.

Q80. There are 100 students in a college class of which 36 are boys studying Statistics and 13 girls are not studying statistics. If there are 55 girls in all, find the probability that a boy picked up at random is not studying Statistics.

Q81. In an examination, 30% of the students have failed in Mathematics, 20% of the students have failed in Chemistry and 10% have failed in both Mathematics and Chemistry. A student is selected at random.

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i) What is the probability that the student has failed in Mathematics if it known that he has failed in Chemistry?

ii) What is the probability that the student has failed either in Mathematics or in Chemistry?

Q82. A bag contains 3 red and 2 white balls. A second bag contains 2 red and 4 white balls. One ball is selected at random from the first bag and transferred to the second bag. Then a ball is drawn at random from the second bag. Find the probability that it is a red ball. Ans: 13/35

Q83. A purse contains 4 five rupee coins and 3 ten rupee coins. Another purse contains 2 five rupee coins and 4 ten rupee coins. If a coin is selected at random from one of the two purses, find the probability that it is a five rupee coin.

Q84. The probability that a student passes in English is 0.8 and the probability that he passes in mathematics given that he has passed in English is 0.6. find the probability that he passes in both the subjects. Ans: 0.48

Q85. The odds that a book will be favourably reviewed by three independent critics are 3 to 2, 4 to 3 and 2 to 3 respectively. What is the probability that of the three reviews a majority will be favorable?

Q86. The odds against A solving a problem are 8 to 6 and odds in favour of B solving the same problem is 14 to 16. If both of the W them try the problem find the probability that. I)The problem will be solved. ii)both A and B will solve the problem.

Q87. The odds in favour of A running a track of 100 meters are 5:3.The odds against B running the same track are 4:3.If both of them start what is the probability that i)both will run the full track ii)only one completes the run. iii)none completes the run?

Q88. The odds in favour of Ashok getting a scholarship for further studies in U.S.A. are 7:5.The odds in favour of Vikas getting a scholarship are 9:7.Find the probability that i)both of them get the scholarship. ii)only one gets the scholarship.

Q89. An investment consultant predicts that the odds against the price of a certain stock will go up during the next week are 2:1 and the odds in favor of the price remaining the same are 1:3. What is the probability that the price of the stock will go down during the next week?

Q90. A speaks truth in 60% and B in 75% of the cases. In what pre4centage of cases are they likely to contradict each other in stating the same fact?

Q91. Among the examinees in an examination 30%, 35% and 45% failed in statistics, in Mathematics and in atleast one of the subjects respectively. An examinee is selected at random. Find the probabilities that (i) he failed I Mathematics only, (ii) he passed in statistics if it is known that he failed in Mathematics.

Q92. A company has four production sections viz. S1,S2,S3,S4 which contribute 30%, 20%, 28% and 22% respectively, to the total output. It was observed that these sections respectively produced 1%,2%,3%and 4% defective units. If a unit is selected at random and found to be defective, what is the probability that the unit so selected has come from either section one or section four

Q93. A company has two plants to manufacture scooters Plant I manufacturers 80% of the scooters and plant II manufacturers 20%.At Plant I, 85 out of 100 scooters are rated standard quality or better. At plant II, only 65 out of 100 scooters are rated standard quality or better.i) What is the probability that the scooter selected at random came

from plant I if it is known that the scooter is of standard quality?

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ii) What is the probability that the scooter came from plant II if it is known that the scooter is of standard quality?

Q94. X Can solve 80% of the problems while Y can solve 90% of the problems given in a statistics book. A problem is selected at random. What is the probability that at least one of them will solve the same?

Q95. Out of the numbers 1 to 120, one is selected at random. What is the probability that it is divisible by 8 or 10?

Q96. Out of five players (of which two are members of a certain club), three are to be selected to represent the country at an international tournament. Find the probability that less than two of those selected to represent are members of the club.

Q97. Three horses A,B and C are in a race. A is twice as likely to win as B, and B is twice as likely to win as C. What are the respective probabilities of winning?

Q98. There are four hotels in a certain town. If 3 men check into hotels in a day, what is the probability they check into a different hotel?

Q99. A committee of 4 people is to be appointed from 3 officers of the production department, 4 officers of the purchase department, two officers of the sales department and one chartered accountant. Find the probability of forming the committee in the following manner:i) There must be one from each category.ii) It should have at least one from the purchase department.iii) The chartered accountant must be in the committee.

Q100. A box contains 4 defective and 6 goods electronic calculators. Two calculators are drawn out one by one without replacement -i) What is the probability that the two calculators so drawn are good?ii) One of the two calculators so drawn is tested and found to be good.

What is the probability that the other one is also good?

Q101. An electronic manufacturer has two lines A and B assembling identical electronic units. The units assembled on line A are 5% defective while those assembled on line B are 10% defective. All defective units must be reworked at a significant increase in cost. During the last 8 hours shift, line A produced 200 units while line B produced 300 units. One unit is selected at random from among the 500 units produced and it is found to be defective. What is the probability that it was assembled (i) on line A? ii) on line B?

Unit 4 - Set theory

Q102. For two events A and B, P (B) = 0.3 P (A but not B) = 0.4 and P (not A) = 0.6. The events A and B are(a) exhaustive (b) independent(c) equally likely (d) mutually exclusive

Q103. Given that for two events A and B, P (A) = 3/5, P (B) = 2/3 and P (A) =3/4, what is P (A/B)?(a) 0.655 (b) 13/60 (c) 31/60 (d) 0.775

Q104. For two independent events A and B, what is P (A+B), given P(A)= 3/5 and P (B) = 2/3?(a) 11/15 (b) 13/15 (c) 7/15 (d) 0.65

Q105. If P (A) = P and P (B) = q, then(a) P (A/B) p/q (b) P (A/B) p/q(c) P (A/B) q/p (d) None of these

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Q106. If P ( ) = 5/6, P (A)= ½ and P = 2/3,, what is P (A B)?(a) 1/3 (b) 5/6 (c) 2/3 (d) 4/9

Q107. If for two independent events A and B, P (A B) = 2/3 and P (A) = 2/5, what is P (B)?(a) 4/15 (b) 4/9 (c) 5/9 (d) 7/15

Q108. If P (A) = 2/3, P(B) = 3/4 , P (A/B) = 2/3, then what is P (B/A)?(a) 1/3 (b) 2/3 (c) 3/4 (d) 1/2

Q109. If P (A) = a, P (B) = b and P (P (A B) = c then the expression of P (A‛B‛) in terms of a, b and c is(a) 1- a – b – c (b) a + b – c(c) 1 +a – b – c (d) 1 –a – b +c

Q110. For three events A, B and C, the probability that only A occur is(a) P (A) (b) P (A B C)(c) P (A’ B C) (d) P (A B’ C’)

Q111. For a group of students, 30%, 40% and 50% failed in Physics, Chemistry and at least one of the two subjects respectively. If an examinee is selected at random, what is the probability that he passed in Physics if it is known that he failed in Chemistry?(a) 1/2 (b) 1/3 (c) 1/4 (d) 1/6

Q112. Given that P (A) = 1/2, P (B) = 1/3, P (A B) = 1/4, what is P (A‛/B‛)(a) 1/2 (b) 7/8 (c) 5/8 (d) 2/3

Q113. Four digits 1,2,4 and 6 are selected at random to form a four digit number. What is the probability that the number so formed, would be divisible by 4?(a) 1/2 (b) 1/5 (c) 1/4 (d) 1/3

Unit V Problems on expected ValuesQ114. If the probability that a man wins a prize of Rs.10 is 3/5 and the

probability that he wins nothing is 2/5 find the mathematical expectation.

Q115. A box contains 3 red,4 green,2 black and 1 white marbles. A man is blindfolded, asked to select a marble. If he selects a red marble he gets Rs 3, for a green one, he wins Rs,2 for a black one,Rs.7 and for a white one Rs.10.What is his mathematical expectation ?

Q116. A die is tossed twice. If it shows the same number twice, Gopal gets Rs.100 otherwise he losses Rs 5.What is his mathematical expectation?

Q117. If a man purchases a raffle ticket he can win a first prize of Rs.5,000 or a second prize of Rs.2,000 with probabilities 0.001 and 0.003.What should be a fair price to pay for the ticket.?

Q118. A bag contains 2 white balls and 3 black balls. Four persons A,B,C,D in that order each draws one ball and does not replace it. The first to draw a white ball receives Rs.10.Determine their expectations.

Q119. In a business venture a man can make a profit of Rs.2,000 with a probability of 0.4 or have a loss of Rs.1000 with a probability of 0.6.What is his expected profit?

Q120. A person tosses two coins simultaneously. He receives Rs.8 for two heads, Rs.2 for one head and he is pay Rs.6 for no head. Find his expectation.

Q121. A dice is loaded in such a way that each odd number is twice as likely to occur as each even number. Find (i) the probability that the number rolled is a perfect square and (ii) the probability that the number rolled is a perfect square provided it is greater than 3.

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Q122. In the play of two dice, the thrower loses if his first throw is 2,4 or 12 he wins if his first throw is a 5 or 11. Find the ratio between his probability of loosing and probability of winning in the first throw.

Q123. (a) A sample of 3 items is selected at random from a box containing 12 items of which 3 are defective. Find the possible number of defective combinations of the said 3 selected items along with probability of a defective combination.

Q124. A box contain 6 tickets. Two of the tickets carry a prize of Rs.5 each, the other four prizes are of Re.1.If one ticket is drawn, what is the expected value of the prize?

Unit 5 objectivesQ1. A packet of 10 electronic components is known to include 3 defectives. If

4 components are selected from the packet at random, what is the expected value of the number of defective?(a) 1.20 (b) 1.21 (c) 1.69 (d) 1.72

Q2. The probability that there is at least one error in an account statement prepared by 3 persons A, B and C are 0.2, 0.3 and 0.1 respectively. If A, B and C prepare 60, 70 and 90 such statements, then the expected number of correct statements.(a) 170 (b) 176 (c) 178 (d) 180

Q3. A bag contains 6 white and 4 red balls. If a person draws 2 balls and receives Rs.10 and Rs.20 for a white and red balls respectively, then his expected amount is(a) Rs.25 (b) Rs.26 c) Rs.29 (d) Rs.28

Miscellaneous Sums. Unit 6

Q125. Two urns contain respectively 3 white, 7 red, 15 black balls and 10 white, 6 red, 9 black balls. One ball is taken from each urn. What is the probability that both will of the same colour?

Q126. Two boxes contain respectively 6 brown, 8 blue, one black balls and 3 brown, 7 blue and 5 black balls. One ball is drawn from each box. What is the probability that both the balls drawn are of the same colour?

Q127. A class consists of 100 students, 25 of them are girls and 75 boys, 20 of them are rich and remaining poor, 40 of them are fair complexioned. What is the probability of selecting a fair complexioned rich girl?

Q128. A candidate is selected for interviews for three posts. For the first post there were 3 candidates, for the second 4 and for the third 2. What is the probability that the candidate is selected for atleast one post?

Q129. 8 boys and 2 girls are to be seated in a row for a photograph. If the arrangement is made at random, find the probability that both the girls will be together.

Q130. Six boys and three girls are to be seated at random in a row for a photograph. Find the probability that no two girls are together.

Q131. Four letters of the word `FAILURE’ are arranged in all possible ways. Find the probability that the word formed is FAIR.

Q132. You have been offered the chance to play a dice game in which you will receive Rs 20 each time the point total of a toss of two dice is 6. If it costs you Rs 2.50 per toss to participate, should you play or not? Will it make any difference in your decision if it costs Rs.3.00 per toss instead of Rs 2.50?

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Q133. What is the probability that over a two day period the number of requests would either be 11 or 12 if at a motor garage the records of service requests along with their probabilities are as given below:

Daily Demand Probability 5 0.25 6 0.65

7 0.10

Q134. The probability that there is at least one error in an account statement prepared by A is 0.3 and for B and C, they are 0.4 and 0.45 respectively. A, B and C prepared 20, 10,40 statements respectively. Find the expected number of correct statements in all.

Q135. Presuming the daily demand to be independent, you are to find the probability that over a two day period the number of requests at some service station will be (i) 9 and (ii) 10 if the past records indicate that daily demand has either been 4 with probability 0.4 or 5 with probability 0.6.

EXERCISE 4.1

Q1. A bag contains 5 white balls, 6 red balls and 7 blue balls. A ball is drawn at random. What is the probability that it is white? a) 5/18 b) 13/18 c) 1 d) 1/18

Q2. A bag contains 5 white balls, 6 red balls and 7 blue balls. Two balls are drawn at random. What is the probability that both balls are white? a) 5/18 b) 1/9 c) 10/153 d) 13/18

Q3. A bag contains 5 white balls, 6 red balls and 7 blue balls. Four balls are drawn at random. What is the probability that two are red and the other two are blue? a) 7/68 b) 2/9 c) 9/21 d) 13/21

Based on the following information answer the questions 4 and 5An employer categorizes job applicants according to whether they have University degree and whether they had relevant work experience. In a large group of applicants, 70 percent have a degree with or without any work experience and 60 percent have work experience with or without a degree. 50 percent of the applicants have both a degree and relevant work experience.

Q4. What is the probability that a randomly selected job applicant has either a degree or relevant work experience? a) 0.4 b) 0.7 c) 0.8 d) 0.2

Q5. What is the probability that the applicant has neither a degree nor work experience? a) 0.7 b) 0.2 c) 0.8 d) 0.5

Q6. Medical records show that one out of ten individuals in a certain town has a low thyroid condition. If 20 persons in this town are randomly chosen and tested, what is the probability that at least one of them will have a low thyroid condition? a) 0.8784 b) 0.1216 c) 0.05 d) 0.9

Based on the following information answer the questions 7 and 8A bag contains two green balls and four red balls and a second bag contains four green balls and three red balls.

Q7. If a balls is drawn at random from one of the two bags, what is the probability that is a green ball? a) 4/7 b) 3/7 c) 4/14 d) 19/42

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Q8. A ball is drawn at random from one of the bag and it turns out to be a green one. What is the probability that it came from the first bag? a) 7/19 b) 19/42 c) 1/7 d) 2/7

Q9. In a firm, 40 percent of the work forces are female, 25 percent of the female workers are management grade and 30 percent of the male workers are management grade. If a management grade worker is selected at random from this firm, what is the probability the worker will be a female? a) 1/14 b) 9/14 c) 5/14 d) 3/5

Q10. In a certain firm, when a worker arrives late there is a one in four chance that he will be caught by the foreman. On the first occasion he is caught, he is given a warning; the second time he is dismissed. What is the probability that a worker who is late three times is not dismissed?a) 27/64 b) 27/32 c) 9/64 d) 27/128

Q11. In a certain city 25 percent of all cars emit excessive amounts of pollutants. If the probability is 0.99, that a car emitting excessive amounts of pollutants will fail a vehicular emission test and the probability is 0.17 that a car not emitting excessive amounts of pollutants will nevertheless fail the test, what is the probability that a car which fails the test actually emits excessive amounts of pollutants?a) 0.99 b) 0.17 c) 0.66 d) 0.34

Q12. If three person, selected at random, are stopped on a street, what are the probabilities that all were born on a Friday? a) 1/7 b) 1/49 c) 1/343 d) 6/7

Q13. In the questions given in 16, what is the probability that two were born on a Friday and the other on a Tuesday? a) 3/7 b)4/7 c) 3/343 d) 1/343

Based on the following data answer the questions 14 to 16.A University’s library has been randomly surveying members over the last month to see who is using the library and what services they have been using. Members are classified a undergraduate, graduate or faculty. Services are classified as reference, periodicals, or books. The data for 350 people are given below. Assume a members uses only one service per visit.

Member Reference Periodical BooksUndergraduate 44 26 72Graduate 24 61 20Faculty 16 69 18Total 84 156 110

Q14. Find the probability that a randomly chosen member is a graduate student.a) 21/70 b) 49/70 c) 12/175 d) 61/105

Q15. Find the probability that a randomly chosen member visited the periodicals section, given the member is a graduate student.a) 61/105 b) 49/70 c) 12/175 d) 61/350

Q16. Find the probability that a randomly chosen member is a faculty member given a reference section visit. a) 5/21 b) 4/21 c) 2/21 d) 4/84

Based on the following data answer the questions 17 and 18At an electronics plant, it is known from past experience that the probability is 0.84 that a new worker who has attended the company’s training program will meet the production quota, and that the corresponding probability is 0.49 for a new worker who has not attended the company’s training program. If 70 percent of all new workers attend the training program,

Q17. What is the probability that a new worker will meet the production quota?a) 0.84 b) 0.49 c) 0.16 d) 0.735

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Q18. If a new worker has not met the production quota, what is the probability that he has attended the training program? a) 0.16 b) 0.112 c) 0.51 d) 0.4226

Q19. In a study of the number of men and women employed at a plant, data shows that 65 percent of the employees are males, 40 percent of the employees are production workers, and the probability that an employee is a male production worker is 0.30. If a randomly selected employee turns out to be a production worker, what is the probability that the employee is a male?a) 0.35 b)0.60 c) 0.70 d) 0.75

Q20. Suppose you are considering the purchase of stock in IJK corporation. You feel that if the sensex rises next year, there is an 85 percent chance that IJK stock will go up. You also feel that there is a 70 percent chance that the sensex will rise next year. What is the probability that both the sensex and the price of IJK will rise next year? a) 0.15 b) 0.30 c) 0.595 d) 0.912

Q21. A box of fuses contains 20 fuses, of which five are defective. If fuses are drawn one at a time from the box and are not replaced, what is the probability that three draws will result in three defective fuses? a) 1/114 b) 113/114 c) 15/18 d) 15/19

Q22. In the question no. 25 if the fuses are drawn one at a time and replaced immediately, what is the probability that the three draws will result in three defective fuses? (a) 5/20 b) 3/18 c) 4/19 d) 1/64

Q23. Football teams from Universities A, B, C and D are quoted as having probabilities of 0.2, 0.4, 0.3 and 0.1 respectively, of lifting the trophy in an inter-university tournament. If, for some reason, University B decides not to participate in the tournament what is the probability that University A will lift the trophy? a) 1/3 b)2/3 c) 9/10 d) 1/10

Q24. A car finance company has three branches in South India i.e., one each at Chennai, Hyderabad and Cochin. The Chennai branch contributes 50% of the total business whereas Hyderabad and Cochin contribute 30% and 20% of the business respectively. It has been observed that 2% of car loans at Chennai and Hyderabad turn into bad debt whereas the 3% loans turned into bad debt in Cochin. Now If a particular car loan has turned into bad debt, what is the probability that it belongs to Chennai branch? a) 0.454 b) 0.273 c) 0.445 d) 0.50

Based on the following information answer the questions 25 and 26.ABC Corporation has submitted a bid for a turnkey project for a 500 MW power plant. If XYZ Corporation, the main competitor of ABC submits the bid, the chances of bid being awarded to ABC are 30%. If XYZ does not bid, there are 75% chances of ABC getting the contact. If there are 50% chances of XYZ bidding for the contract.

Q25. What is the probability of ABC getting the contract? a) 0.2817 b) 0.525 c) 0.15 d) 0.375

Q26. What is the probability that XYZ has submitted the bid, if ABC gets the contract? a) 0.2857 b) 0.15 c) 0.85 d) 0.625

Q27. There are two security analysts A and B. The analysis of A has been found to be accurate in 75% of the cases and the analysis of B is found to be accurate in 80% of the cases. If a particular security is given to them for analysis, what is the probability that they will contradict each other? a) 20% b) 25% c) 35% d) 65%

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Q28. A portfolio manager A has been able to beat the market in four out of 5 periods, portfolio manager B has been able to beat the market in 3 out of four periods and portfolio manager C has been able to beat the market in 2 out of three periods. Assuming that the past performance of all the three portfolio manager will be repeated in future also, what is the probability of all the portfolio manager A, B and C beating the market index.a) 1/5 b) 2/5 c) 4/5 d) 3/5

Q29. There are two bags A and B. The bag A contains three black and two white balls whereas bag B contains two black and three white balls. One ball at random is transferred from bag A to B and then two balls at random are picked from the bag B one by one. You are required to find out the probability of the second ball to be black with replacement of the first ball. a) 13/30 b) 2/15 c) 2/5 d) 1/15

Q30. X and Y are two team leaders in a department in an organisation. Of all the project proposals reaching the department head, 60% are proposed by X and 40% are proposed by Y. A proposal given by X has a 40% chance of being approved and a proposal given by Y has a 55% chance of being approved by the department head. What is the probability of a project being approved?a) 0.24 b) 0.22 c) 0.46 d) 0.25

Based on the following data answer the questions 31 and 32A purchasing agent for trucking firm is considering to change the brand of tyres he buys. To test two other brands, he purchases 25 tyres of each and places them on randomly selected trucks, and measures miles to first recap. The frequency distributions for the two brands of tyres are given below:

Miles to first recap (‘000) Brand A Brand B48-5152-5556-5960-6364-6768-71

243871

457630

25 25

Q31. What is the probability that a tyre selected at random from the group of Brand B lasts between 52,000 and 56,000 miles?a) 1/5 b) 5/8 c) 9/50 d) 2/5

Q32. What is the probability that a tyre selected at random from the group of Brand A lasts between 60,000 and 64,000 miles?a) 8/25 b) 6/25 c) 2/5 d) 3/5

Based on the following information answer the questions 33 and 34.J.J. Construction Co. Ltd., is involved in the construction business. The General Manager of the firm has come to know about the likelihood of a strike by some of its employees. It is understood from the information available that there is a 60% probability that its semi-skilled workers will go on strike and 80% probability that its unskilled workers will go on strike. Further, if the unskilled workers go on a strike there is a 72% probability that its semi-skilled workers will go on a strike. A strike by semi-skilled workers and a strike by unskilled workers are dependent events.

Q33. What is the probability that both the groups will go on strike together?a) 22.4% b) 57.6% c) 96% d) 4%

Q34. What is the probability that the unskilled workers will go on a strike but the semi-skilled workers will not go on a strike? a) 96% b) 4% c) 24% d) 22.4%

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Q35. A and B are two non-mutually exclusive and independent events. If P(A)=0.60 and P(B)=0.50 then, what is the probability of either event A or event B taking place? a) 0.10 b) 0.30 c) 0.50 d) 0.80

Q36. In a survey of 100 readers, it was found that 50 red magazine A, 30 read magazine B and 15 read both the magazines. What is the probability of finding a person in the group who reads neither magazine A nor magazine B?a) 0.15 b) 0.35 c) 0.65 d) 0.80

Q37. A firm manufactures steel pipes in three plants viz, A, B and C. The daily production volumes from the three firms A, B and C respectively are 1000 units 2000 units and 4000 units respectively. It is known from past experience that 2% of the output from plant A, 3% of the output B and 5% of the output from C are defective. A pipe is selected from a days total production and found to be defective. What is the probability that the pipe is manufactured by plant B? a) 14.29% b) 21.43% c) 28.57% d) 4%

Q38. Three candidates A, B and C are being considered by the Board of Directors of a company for the post of the CEO. The probability that A will be appointed as the CEO is 0.20, the probability that B will be appointed is 0.30 and the probability that c will be appointed to the post is 0.50. If A is appointed as the CEO then the probability of successfully launching a new product is 30%. If B is appointed as the CEO then the probability of successfully launching a new product is 50% and if C is appointed then the probability of successfully launching a new product is 60%.Find the probability of successful launching of the new product.a) 6% b) 15% c) 30% d) 51%

Q39. A, B and C are mutually exclusive and collectively exhaustive events. Resulting from an experiment. B is twice as likely as A, and C is 2.5 times as likely as B. a) P(A) = 3%, P(B) = 6%, P(C) = 15%b) P(A) = 5%, P(B) = 10%, P(C) = 25%c) P(A) = 12.5%, P(B) = 25%, P(C) = 62.5%d) P(A) = 8%, P(B) = 16%, P(C) = 40%

Q40. Experiment 1 results in the mutually exclusive and collectively exhaustive events A, B and C. Experiment 2 results in the mutually exclusive and collectively exhaustive events D and E. The events resulting from experiment 1 are independent of the events resulting from experiment 2 and vice versa. The joint probabilities of the events that may result from these two experiments are given below:

A B CD 0.30 0.18 ?E 0.20 0.12 0.08

What is the joint probability of the events C and D occurring?a) 12% b) 20% c) 32% d) 40%

Q41. Two fair dice are thrown. What is the probability that one of them gives an even number less than 5, and the other one gives an odd number less than 4? a) 1/9 b) 2/9 c) 1/3 d) 4/9

Q42. A box contains one green ball and three blue balls. A second box contains two green balls and four blue balls. A third box contains three balls and one blue ball. One of the three boxes is selected at random and green a ball is randomly taken out of it. What is the likelihood that the ball is green? a) 5/12 b) 1/9 c) 4/9 d) 1/4

Q43. In a certain part of a city, the number of residents is 6000. out of the 6,000 residents, 1,440 are above 30 years of age, and 3,600 are females. Out of the 1,440 residents who are above 30 years of age, 240 are females.

157


What is the probability that a resident selected at random is either a female or over 30 years of age? a) 0.04 b) 0.24 c) 0.60 d) 0.80

Q44. From twenty tickets marked with the first twenty numerals, one ticket is drawn at random. What is the probability that the numeral marked on it is a multiple of 3 or of 5? a) 1/20 b) 1/10 c) 3/10 d) 9/20

Q45. The ratio of the likelihood that an individual A, who is 35 years old now, will live up to an age of 65, to the likelihood that he will die before 65 is 7:9. The ratio of the likelihood that another individual B, who is 45 years old now, will live up to an age of 75, to the likelihood that he will die before 75 is 2:3What is the likelihood that at least one of these individuals will be alive 30 years hence?a) 9/16 b) 3/5 c) 27/80 d) 53/80

Q46. Experiment 1 results in the mutually exclusive and collectively exhaustive events A, B and C. Experiment 2 results in the mutually exclusive and collectively exhaustive event D and E. The joint probabilities of the events that may result from these two experiments are given below:

A B CD 0.12 0.075 0.12E 0.28 0.225 0.18

Which of the following statements are correct?i) P(A) = 0.30 ii) P(B) = 0.40iii) P(C) = 0.30 iv) P(D) = 0.685v) P(E) = 0.685 vi) P(D|C) = 0.25vii) P(E|B)= 0.75 viii) P(D|A) = 0.30a) i, iv and vib) iv, vi, vii c) ii, iii, iv vi d) iii, v, vii, viii

ANSWERS 4.1

1.(a) 2.(c) 3.(a) 4.(c) 5.(b) 6.(a) 7.(d) 8.(a) 9.(c) 10.(b) 11.(c) 12.(c) 13.(c) 14.(a) 15.(a) 16.(b) 17.(d) 18.(d) 19.(d) 20.(c) 21.(a) 22.(d) 23.(a) 24.(a) 25.(b) 26.(a) 27.(c) 28.(b) 29.(a) 30.(c) 31.(a) 32.(a) 33.(b) 34.(d) 35.(d) 36.(b) 37.(b) 38.(d) 39.(c) 40.(a) 41.(a) 42.(c) 43.(d) 44.(d) 45.(d) 46.(d)

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EXERCISE 4.2

Q1. A die is thrown once. Then the probability of getting a number greater than 3 is

(a) (b) (c) 6 (d) 0

Q2. Two dice are thrown. The number of sample points in the sample space when six does not appear on any one side is.(a) 11 (b) 18 (c) 30 (d) 25

Q3. A dice is tossed twice. The probability of having a number greater than 4 on each toss is

(a) (b) (c) (d) (e)

Q4. Two Fair dice are tossed. The probability that both shown an even number is

(a) (b) (c) (d)

Q5. Two uniform dices marked 1 to 6 are tossed together. The probability of the total 7 in a single throw is

(a) (b) (c) (d)

Q6. A balanced die is rolled twice. The probability that the sum of the figure observed is 6 is equal to

(a) (b) (c) (d)

Q7. In a single throw of a two dice, the chance of throwing a sum 8 is

(a) (b) (c) (d)

Q8. In a single throw of a pair of dice, P(a total>12) is equal to,

(a) (b) 1 (c) 0.89 (d) 0

Q9. Two dice are thrown simultaneously. The probability of getting a pair of aces is

(a) (b) (c) (d) None

Q10. Two unbiased dice are rolled. The probability that both the dice show the same digit is

(a) (b) (c) (d)

Q11. Two dice are thrown simultaneously. The probability that exactly one die shows 4 is

(a) (b) (c) (d)

Q12. In a single throw of a two dice, the probability that the sum of the scores is at least 10 is

(a) (b) (c) (d)

Q13. Two unbiased dice are thrown. The probability that the product of numbers on their upper faces is 12 is

159


(a) (b) (c) (d)

Q14. Two dice are rolled. The probability that the score on the second die is greater than the score on the first die is

(a) (b) (c) (d)

Q15. Two dice are thrown. The probability that the sum of the scores on their upper face is multiple of 6 is

(a) (b) (c) (d)

Q16. Two unbiased dice are rolled, the probability that the sum of the points on their upper faces is divisible by 4 is

(a) (b) (c) (d)

Q17. Two uniform dice marked 1 to 6 are tossed together. The probability of the total is prime number in a single throw is

(a) (b) (c) (d)

Q18. Two dice are rolled. The probability that atleast one die shows 5 is

(a) (b) (c) (d)

Q19. Two unbiased dice are rolled. The probability that the sum of the points on their upper face is a perfect square is

(a) (b) (c) (d)

Q20. Two dice are thrown. The probability that the sum of square of the digits on their upper faces is 25 is

(a) (b) (c) (d)

Q21. Two dice one green and the other red are rolled and separate scores recorded. The probability that scores on the dice differ by not more than 2 is

(a) (b) (c) (c)

Q22. A fair die is tossed twice. The probability of getting a 4,5 or 6 on the first toss and 1,2,3,4 on the second toss is

(a) (b) (c) (d)

Q23. A perfect cubic die is thrown. The probability that a perfect square comes up is

(a) (b) (c) (d)

Q24. An unbiased die is thrown. The probability of getting an odd number is

(a) (b) (c) (d)

Q25. Three identical dice are rolled. The probability that same number will appear on each of them is

(a) (b) (c) (d)

Q26. If three dice are thrown simultaneously then the probability of getting a sum of score of 5 is

(a) (b) (c) (d)

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Q27. If two dice are rolled the probability of getting multiple of 2 on the first and multiple of 3 on the second is

(a) (b) (c) (d)

Q28. If two dice are rolled, the probability of getting a total of 10 is

(a) (b) (c) (d)

Q29. If a coin is tossed, the probability of getting either head or tail is

(a) 1 (b) (c) (d)

Q30. If two coins are tossed, then the probability of getting exactly one head is

(a) (b) (c) (d) 1

Q31. Two coins are tossed. The probability of getting atleast one head is(a) 1/2 (b) 1/4 (c) 3/4 (d) 1

Q32. Two coins are tossed. The probability of getting no head is(a) 3/4 (b) 1/4 (c) 1/2 (d) 1

Q33. Two unbiased coins are tossed. The probability of getting the same face on both is(a) 1/4 (b) 1/2 (c) 3/4 (d) 1

Q34. Three unbiased coins are toss. The probability of getting at least two heads up is(a) 7/8 (b) 3/8 (c) 1/2 (d) 1/4

Q35. A coin is tossed three times. The probability of getting at least one head up is(a) 7/8 (b) 1/2 (c) 3/8 (d) 1/4

Q36. A coin is tossed three time. The probability of getting the second is not a head(a) 3/8 (b) 1/4 (c) 5/8 (d) 1/2

Q37. Three coins are tossed once. The probability of getting atmost 2 head is(a) 3/8 (b) 1/2 (c) 7/8 (d) 1/2

Q38. When the three coins are tossed simultaneously then the probability of getting one head will be(a) 7/8 (b) 3/8 (c) 1/7 (d) 3/7

Q39. Ram Lal throws three coins. The probability of atmost one tail turning up is(a) 1/4 (b) 1/2 (c) 3/8 (d) 5/8

Q40. Thee unbiased coins are tossed. The probability of getting exactly two tails is(a) 1/8 (b) 1/2 (c) 7/8 (d) 3/8

Q41. Three unbiased coins are tossed. The probability of getting the same face on all three coins is(a) 1/8 (b) 1/4 (c) 3/8 (d) 7/8

Q42. A fair coin is tossed 100 times. The probability of getting tails an odd number of times is(a) 1/2 (b) 1/8 (c) 3/8 (d) none

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Q43. A coin is tossed once. If a head comes up then it is tossed again and if a tail comes up, a dice is thrown. The number of point in the sample space of experiment is(a) 24 (b) 12 (c) 4 (d) 8

Q44. There are 15 tickets bearing the number from 1 to 15 in a bag and one ticket is drawn at random from the bag. The probability of the events of getting a ticket bearing a number which is a multiple of 5 is(a) 7/15 (b) 1/5 (c) 2/15 (d) 1/3

Q45. There are 20 tickets numbered from 1 to 20. One ticket is drawn at random. The probability that the ticket bears a number which is divisible by 3 is(a) 1/10 (b) 3/10 (c) 3/20 (d) 7/20

Q46. There are 20 ticket numbered from 1 to 20. One ticket is drawn at random. The probability that the ticket bears a number which is a perfect square is(a) 1/5 (b) 3/20 (c) 1/4 (d) 1/10

Q47. a card is drawn at random from a well shuffled pack of 52 cards. The probability that a red card drawn is(a) 1/4 (b) 1/26 (c) 1/2 (d) 1/13

Q48. a card is drawn at random from a well shuffled pack of 52 cards. The probability that a king card drawn is(a) 1/26 (b) 1/13 (c) 1/4 (d) 1/2

Q49. A card is drawn at random from a well shuffled pack of 52 cards. The probability that a face card drawn is(a) 2/13 (b) 3/13 (c) 1/4 (d) 1/26

Q50. A card is drawn at random from a well shuffled pack of 52 cards. The probability that a spade card drawn is(a) 1/4 (b) 3/13 (c) 1/52 (d) 1/26

Q51. A card is drawn at random from a well shuffled pack of 52 card. The probability that the card drawn is bears a number between 4 and 7 both inclusive is(a) 1/13 (b) 1/26 (c) 4/13 (d) 1/4

Q52. A card is drawn from a well shuffled pack of 52 cards. The probability that the card drawn is an ace is(a) 3/13 (b) 1/13 (c) 5/26 (d) 1/52

Q53. A card is drawn from a pack of well shuffled 52 playing cards. The probability that the card drawn is bearing a number between and including 2 and 6 is(a) 3/13 (b) 5/13 (c) 5/26 (d) 19/52

Q54. A card is drawn at random from a well shuffled pack of 52 card. The probability that the card drawn is bears a number between 3 and 8 both exclusive is(a) 5/13 (b) 4/13 (c) 1/26 (d) 1/4

Q55. A card is drawn at random from a well shuffled pack of 52 cards. The probability that the card drawn is a black queen is(a) 3/13 (b) 1/13 (c) 1/26 (d) 3/52

Q56. From a pack of well-shuffled 52 cards one card is drawn at random. The probability that the card drawn is a jack or a king(a) 1/13 (b) 3/13 (c) 2/13 (d) 1/4

Q57. From a pack of well shuffled 52 cards, one card is drawn at random. The probability that the card drawn is a king or a card of spade is(a) 4/13 (b) 17/52 (c) 3/13 (d) 15/52

162


Q58. A card is drawn at random from a pack 52 cards. The probability that the card drawn is a diamond or a spade is(a) 1/52 (b) 1/4 (c) 4/13 (d) 1/2

Q59. A card is drawn from a pack 52 cards. The probability that a the card drawn is a red queen or a black king is(a) 4/13 (b) 1/26 (c) 1/13 (d) 2/13

Q60. A card is drawn at random from a pack of 52 cards. The probability that the card drawn is an ace or a black queen or a king of heart is(a) 7/52 (b) 4/13 (c) 1/26 (d) 2/13

Q61. Two cards are drawn at random from a pack of 52 cards. The probability that both are face cards is(a) 8/663 (b) 13/102 (c) 11/221 (d) 1/17

Q62. Two cards are drawn at random from a pack of 52 cards. The probability that both are black is(a) 8/663 (b) 11/221 (c) 1/17 (d) 25/102

Q63. From a well-shuffled pack of 52 card, two cards are drawn at random. The probability that both the cards are diamonds is(a) 1/17 (b) 11/221 (c) 8/663 (d) 1/26

Q64. From a well-shuffled pack of 52 cards two cards are drawn at random. The probability that one is king and the other is a queen is(a) 1/26 (b) 8/663 (c) 25/102 (d) 1/17

Q65. From a well-shuffled pack of 52 cards, two cards are drawn at random. The probability that one is king and the other is spade.(a) 1/26 (b) 1/17 (c) 8/663 (d) 25/102

Q66. From a pack of well-shuffled 52 cards, two cards are drawn at random. The probability that one is heart and the other is spade is(a) 1/7 (b) 4/663 (c) 13/102 (d) 1/26

Q67. From a pack of well-shuffled 52 cards, two cards are drawn at random. The probability that one is jack and the other an ace is(a) 13/102 (b) 1/7 (c) 8/663 (d) 1/26

Q68. From a pack of 52 cards, two cards are drawn at random. The probability that the two cards drawn contain exactly one face card is(a) 30/221 (b) 19/34 (c) 80/221

Q69. From a pack of 52 cards, two cards are drawn at random. The probability that the two cards drawn contain exactly one ace is(a) 80//221 (b) 188/221 (c) 13/102 (d) 32/221

Q70. From a pack of 52 cards, two cards are drawn at random. The probability that the two cards drawn contain no club is(a) 19/34 (b) 13/102 (c) 1/26 (d) 4/663

Q71. Two cards are drawn at random from a pack of 52 cards. The probability that one is king and one is queen (a) 19/34 (b) 188/221 (c) 8/663 (d) 32/221

Q72. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain two kings and one ace is(a) 1/5525 (b) 6/5525 (c) 2/17 (d) 11/850

Q73. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards are black is(a) 6/5525 (b) 2/17 (c) 11/850 (d) none

Q74. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards are diamond is(a) 2/17 (b) 11/850 (c) 3/11050 (d) 741/1700

163


Q75. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that three cards are queen is(a) 1/5525 (b) 11/850 (c) 2/17 (d) none

Q76. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain exactly one club card is (a) 38/85 (b) 741/1700 (c) 1128/5525 (d) 703/1700

Q77. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain 2 queen and a jack is(a) 2/17 (b) 1/5525 (c) 6/5525 (d) 38/55

Q78 From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain 2 aces and a 10 of diamonds is(a) 3/11050 (b) 741/1700 (c) 38/55 (d) 2/17

Q79. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain no-diamond is(a) 38/55 (b) 6/6625 (c) 703/1700 (d) 741/1700

Q80. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain exactly one jack is(a) 38/55 (b) 1128/5525 (c) 741/1700 (d) 11/850

Q81. From a well-shuffled pack of 52 cards, 3 cards are drawn at random. The probability that the three cards drawn contain no face card is(a) 38/85 (b) 741/1700 (c) 11/850 (d) none

Q82. If three cards are drawn at random from pack of 52 playing cards find the probability that all cards are of the same suit is(a) 4/850 (b) 22/425 (c) 22/85 (d) 11/85

Q83. An urn contain 9 red, 7 white and 4 black balls out of them one is drawn at random. The probability that the ball drawn will be a red or black is(a) 9/20 (b) 7/20 (c) 11/20 (d) 13/20

Q84. A box contains 5 red, 11 white and 7 black balls. One ball is drawn at random. The probability that the ball drawn is a white ball is(a) 13/23 (b) 10/25 (c) 11/23 (d) 12/23

Q85. A bag contains 6 red, 5 Blue, 3 white and 4 black balls. A ball is drawn at random. The probability that the ball is red or black is(a) 5/9 (b) 5/8 (c) 4/9 (d) 3/8

Q86. In a bag there are 6 black, 4 white and 3 yellow balls. A ball is taken at random. The probability of getting a yellow or white ball is(a) 8/13 (b) 6/13 (c) 9/13 (d) 7/13

Q87. A bag contains 7 white balls, 5 black balls, 4 red balls. If two balls are drawn at random from the bag. The probability that one is black and other is red.(a) 1/7 (b) 3/7 (c) 2/7 (d) 1/6

Q88. a box contains 7 red, 5 white and 8 green balls identical in all respects excepts colour. One ball is drawn at random. The probability that it is not white is(a) 3/4 (b) 3/5 (c) 1/4 (d) 1/2

Q89. A box contains 7 white, 5 black and 4 red balls. One ball is drawn at random. The probability that it is white or black or red is(a) 1/4 (b) 3/4 (c) 1 (d) 1/2

Q90. A box contains 3 yellow, 4 blue and 5 white balls. Two balls are drawn at random. The probability that the two balls drawn are blue.(a) 2/11 (b) 1/11 (c) 3/11 (d) 5/22

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Q91. A box contains 3 yellow, 4 blue and 5 white balls. Two balls are drawn at random. The probability that the two balls drawn are not yellow is(a) 1/11 (b) 6/11 (c) 5/11 (d) 5/22

Q92. A box contains 3 yellow, 4 blue and 5 white balls. Two balls are drawn at random. The probability that the two balls drawn consist of a yellow and a white ball is(a) 6/11 (b) 7/22 (c) 5/22 (d) 5/11

Q93. A box contain 3 yellow 4 blue and 5 white balls. Two balls are drawn at random. The probability that the two balls drawn consist of exactly one blue balls is(a) 15/33 (b) 16/33 (c) 2/33 (d) 14/33

Q94. A bag contains 2 black, 3 red and 5 blue balls. The three balls are drawn at random. The probability that the three balls drawn are not black is(a) 1/6 (b) 7/40 (c) 7/15 (d) 1/4

Q95. A bag contains 2 black, 3 red and 5 blue balls. Three balls are drawn at random. The probability that the three balls are of different colour is(a) 7/15 (b) 1/6 (c) 3/4 (d) 1/4

Q96. A bag contains 6 white and 9 black balls. If 3 balls are drawn at random. The probability that all of them are black is(a) 1/5 (b) 7/15 (c) 12/65 (d) 11/65

Q97. In a bag there are 6 white, 4 black and 5 yellow balls. Three balls are, taken at random from the bag. The probability that all are of the same colour is(a) 25/38 (b) 34/455 (c) 11/15 (d) 3/11

Q98. A bag contains 3 white balls, 4 black balls and 5 red balls. If 3 balls are drawn at random. The probability that 2 are red and 1 is black is(a) 1/55 (b) 2/11 (c) 3/11 (d) none

Q99. A bag contains 2 black, 3 red and 5 blue balls. Three balls are drawn at random. The probability that the three balls drawn consist of exactly 2 red balls is(a) 7/15 (b) 5/40 (c) 7/40 (d) 1/4

Q100. A bag contains 3 white balls, 4 black balls and 5 red balls. If 3 balls are drawn at random. The probability that the three balls drawn 1 of each colour is(a) 2/11 (b) 1/11 (c) 3/11 (d) 1/22

Q101. A bag contains 2 black, 3 red and 5 blue balls. Three balls are drawn at random. The probability that the three balls drawn consist of exactly one black ball is(a) 7/40 (b) 1/120 (c) 7/15 (d) 1/4

Q102. From a group of 5 men and 4 women, 4 persons are selected at random to form a committee. The probability that the committee contains 3 men and a women is(a) 22/63 (b) 19/63 (c) 35/286 (d) 20/63

Q103. A committee of 5 is to be formed from a group of 8 boys and 7 girls. The probability that the committee consists of 3 boys and 2 girls is(a) 56/143 (b) 56/144 (c) 1170/3003 (d) None

Q104. From a group of 5 men, 3 women and 6 children. 4 persons are chosen at random. The probability that the group selected contains all children.(a) 30/1001 (b) 31/1001 (c)29/1001 (d)15/1001

Q105. From a group of 5 men and 4 women, 4 persons are selected at random to form a committee. The probability that the committee has at least 3 women is(a) 5/14 (b) 20/63 (c) 1/6 (d) 5/6

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Q106. From a group of 4 men, 4 women and 2 children, 4 persons are selected at random. The probability that the committee contains exactly 2 men are selected is (a) 5/42 (b) 3/7 (c) 2/7 (d) 1/3

Q107. From a group of 5 men and 4 women 4 persons are selected at random to form a committee. The probability that the committee has a majority of men is(a) 1/6 (b) 3/14 (c) 5/14 (d) 20/63

Q108. A class of 40 students contains 10 girls. A committee of 5 students is to be formed at random. The probability that there are exactly 2 girls in the committee is


Q109. From a group of 5 men and 4 women, 4 persons are selected at random to form a committee. The probability that the committee has a majority of women is(a) 5/14 (b) 1/6 (c) 5/13 (d) 1/3

Q110. From a group of 4 men, 4 women and 2 children, 4 persons are selected at random. The probability that no child selected is(a) 2/3 (b) 4/3 (c) 1/3 (d) 3/7

Q111. From a group of 4 men, 3 women and 5 children, 4 persons are selected at random. The probability that the group selected consist of no women is.(a) 14/55 (b) 14/99 (c) 4/495 (d) 13/55

Q112. From a group of 5 men and 3 women a committee of 4 persons is to be selected. If selection is done at random. The probability that the committee contains exactly two persons of either sex is(a) 13/70 (b) 3/7 (c) 2/7 (d) 15/17

Q113. A committee of 4 boys and 3 girls is to be formed from a group of 8 boys and 5 girls selecting randomly. The probability that the committee contains a particular boys and particular girl is(a) 1/10 (b) 2/5 (c) 3/10 (d) 1/2

Q114. A committee of four is to be formed from 10 boys and 1 girl at random. The probability that the girls is included is(a) 3/11 (b) 2/11 (c) 4/11 (d) 5/11

Q115. A team of 5 is to be selected from 8 boys and 3 girls. The probability that it includes 2 particular girls is(a) 4/11 (b) 2/11 (c) 5/11 (d) 3/11

Q116. An organisation consists of 25 members including 4 doctors. A committee of 4 is formed at random. The probability that the committee contains at most 1 doctor is(a) 1197/2530 (b) 17/2530 (c)126/1265 (d) 2261/2530

Q117. A committee of 5 is to be formed from a group of 8 boys and 7 girls. The probability that the committee contains at least one girl is(a) 8/429 (b) 421/429 (c) 423/429 (d) 7/429

Q118. A committee of 5 is to be formed at random from a group of 7 boys and 5 girls. The probability that the committee includes at least one girl is(a) 257/264 (b) 251/264 (c) 255/264 (d) 37/39

Q119. Two cards are drawn at random from a well shuffled pack of 52 cards. The probability that at least one of them is diamond is (a) 15/34 (b) 13/34 (c) 1/2 (d) 7/17

Q120. A room has 3 electric lamps form a collection of 12 electric bulbs of which 6 are good, 3 bulbs are selected at random and put in the lamps. The probability that the room is lighted by at least one of the bulbs is(a) 9/11 (b) 10/11 (c) 3/11 (d) 8/11

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Q121. A room has 4 lamps. From a collection of 15 bulbs of which 8 are defective, 4 are selected at random and put in the sockets. The probability that the room is lighted is (a) 38/39 (b) 37/39 (c) 10/11 (d) 20/39

Q122. In a box containing 10 bulbs 4 are defective. If 3 bulbs are selected at random. The probability that at least one of these three bulbs is non defective is.(a) 29/30 (b) 14/15 (c) 37/39 (d) 2/39 (e) 5/6

Q123. A room has three lamps. From a collection of 10 bulbs of which 6 are defective, a persons selects 3 at random and put them into the lamps. The probability that there is light from all three lamps is(a) 29/30 (b) 37/39 (c) 1/30 (d) 5/6

Q124. A room has three lamps from a collection of 10 bulbs of which 6 are defective, a person selects 3 at random and puts them into the lamps. The probability that there is light from at least one lamps is(a) 29/30 (b) 37/39 (c) 1/30 (d) 5/6

Q125. In a batch of 400 bolts. 50 are found to be defective. A bolt is selected at random from the batch. The probability that it is not-defective is(a) 5/8 (b) 7/8 (c) 3/4 (d) 1

Q126. In a batch of 400 bolts, 20 are defective. The probability that a bolt selected at random is non defective is(a) 1/20 (b) 9/10 (c) 19/20 (d) 1

Q127. A box contains 10 radio valves all apparently sound, although 4 of them are substandard. The probability if two of the valves selected at random are substandard is(a) 1/15 (b) 1/5 (c) 2/15 (d) 4/15

Q128. A lot of 400 articles manufactured in a factory contains 25 defective articles. If two articles are picked up from the lot of at random, the probability that they are non-defective is(a) 935/1064 (b) 129/1064 (c) 934/1064 (d) none

Q129. A box contains 12 screws of which 4 are defective and the others are usable. A persons selects, 3 screws at random from the box. The probability that be gets at least 2 usable screws is(a) 3/55 (b) 41/55 (c) 42/55 (d) 43/55

Q130. A single letters is selected at random from the word. `PROBABILITY’. The probability that is a vowel is(a) 1/11 (b) 4/11 (c) 2/11 (d) 0

Q131. In a box containing 100 bulbs, 10 are defective, what is the probability that out of a sample of 5 bulbs none is defective.

(a) 10-5 (b) (c) (d) 9/10

Q132. A coin is tossed 8 times. The probability of getting a head three times is(a) 7/16 (b) 7/64 (c) 7/128 (d) 7/32

Q133. A coin is tossed n times. The probability of getting atleast one head is greater than that of getting at least two tails by 5/32. Then n is(a) 5 (b) 10 (c) 15 (d) none

Q134. A coin is tossed 4 times. The probability that at least one head turns up is(a) 1/16 (b) 2/16 (c) 14/16 (d) 15/16

Q135. Four unbiased coins are tossed. The probability of getting at least two tails is

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(a) 15/16 (b) 14/16 (c) 11/16 (d) 1/16

Q136. Five unbiased coins are tossed. The probability of getting at least two heads is(a) 31/32 (b) 13/16 (c) 15/16 (d) 30/32

Q137. Five unbiased coins are tossed. The probability of getting at least one tail is(a) 13/16 (b) 15/16 (c) 31/32 (d) none

Q138. Three mangoes and three apples are in box. If two fruits are chosen at random. The probability that one is a mongo and the other is an apple is(a) 2/3 (b) 3/5 (c) 1/3 (d) none

Q139. A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that they are of the same colour is(a) 5/108 (b) 1/6 (c) 5/18 (d) 4/9

Q140. A card is drawn at random from a pack of 100 cards numbered 1 to 100. The probability of drawing a number which is square is(a) 1/5 (b) 1/10 (c) 2/5 (d) none

Q141. One die and one coin are tossed simultaneously. The probability of getting 6 die and head on coin is(a) 1/2 (b) 1/6 (c) 1/12 (d) none

Q142. A coin is tossed and a die is rolled. The chance that the coin shows a head and the die shows 3 is(a) 1/8 (b) 1/12 (c) 1/2 (d) 1

Q143. Three letters are written to difference persons and addresses on three envelopes are also written without looking at the address. The probability that the letters go into right envelops is(a) 1/27 (b) 1/6 (c) 1/9 (d) none

Q144. There are 4 addressed envelopes and 4 letters. Then the chance that all the letters are not mailed through proper envelop is(a) 1/24 (b) 1 (c) 23/24 (d) 7/10

Q145. Of cigarette smoking population 70% are men and 30% are women 10% of these men 20% of these women smoke wills. Probability that a person seen smoking a wills to be male is(a) 1/5 (b) 7/13 (c) 5/13 (d) 5/13

Q146. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three verities is equilateral equals(a) 1/2 (b) 1/5 (c) 1/10 (d) 1/20

Q147. The probability that a leapyer selected at random contains 53 Sunday is(a) 7/336 (b) 26/183 (c) 1/7 (d) 2/7

Q148. The probability that a leap year will have 53 Tuesday is(a) 3/7 (b) 4/7 (c) 2/7 (d) 5/7

Q149. The probability that a leap year will have 52 Mondays is(a) 5/7 (b) 6/7 (c) 1/7 (d) 2/7

Q150. The probability that a non-leap year will have 53 Thursday is (a) 1/7 (b) 5/7 (c) 2/7 (d) 6/7

Q151. The probability that a non-leap year will have 52 Saturday is(a) 2/7 (b) 1/7 (c) 6/7 (d) 5/7

Q152. Ten pairs of shoes are in closet four shoes are selected at random. The probability that there is at least one pair among the four selected is(a) 100/323 (b) 99/323 (c) 98/323 (d) none

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Q153. Out of 13 applicants for a job there are 5 women and 8 men. It is desired to select 2 persons for the job. The probability that at least one of the selected persons will be a woman is(a) 25/39 (b) 14/39 (c) 5/13 (d) 10/13

Q154. Four pairs of hands gloves are there in a closet. Two gloves are drawn at random. The probability that both are for the same hand is(a) 1/7 (b) 2/7 (c) 3/7 (d) 1/14

Q155. Six boys ad three girls are to be seated at random in a row for a photograph. The probability that no two girls are together is(a) 5/12 (b) 7/12 (c) 1/13 (d) ¼

Q156. Five boys and 3 girls are to be seated on chairs in a row. If the arrangement is made at random the probability that no two girls will be seated side by side is(a) 9/14 (b) 1/14 (c) 5/14 (d) 1/2

Q157. 8 boys and 2 girls are sitting in a row for a photograph. If the arrangement is made at random. The probability that both the girl will be together. (a) 1/5 (b) 2/5 (c) 3/5 (d) 5/14

Q158. 8 Indians and 3 Americans are to stands in a row at random. The probability that no two Americans are together is(a) 29/55 (b) 27/55 (c) 1/5 (d) 28/55

Q159. Words from the letters of the word PROBABILITY are formed by taking all at a time. The probability that both B’s are together and I’s are together is(a) 1/55 (b) 2/55 (c) 4/165 (d) none

Q160. From a group of 4 boys and 3 girls candidates are arranged at random, one after the other for an interview. The probability that the boys and girls alternative is.(a) 1/34 (b) 1/35 (c) 1/33 (d) 1/32

Q161. Five Engineering four mathematics, two chemistry books are placed on a table at random. The probability that the books of each kind are all together is(a) 2!5!4!/11! (b) 3!5!4!2!/11!(c) 5!4!2!/10! (d) 3!5!4!2!/10!

Q162. 6 boys and 6 girls sit in a row randomly. The probability that all 6 girls sit together is(a) 1/64 (b) 1/8 (c) 1/132 (d) none

Q163. If different words are found from letters of the word `UNIVERSITY’, then the probability that two I’s don’t come together is(a) 4/5 (b) 2/5 (c) 6/5 (d) 3/5

Q164. The letter of the word `LOGARITHM’ are arranged at random. The probability that, the arrangement start and end with vowels is(a) 3/12 (b) 1/4 (c) 1/12 (d) 5/12

Q165. The letters of the word `LOGARITHM’ are arranged at random. The probability that the arrangements starts with a vowel and ends with a consonant is (a) 1/3 (b) 3/4 (c) 1/4 (d) ½

Q166. The letters of the words `EQUATION’ are arranged at random. The probability that the arrangement starts with a vowel and ends with a consonant is(a) 15/16 (b) 14/56 (c) 16/56 (d) 5/14

Q167. The letters of the word EQUATION are arranged at random. The probability that the arrangement starts and ends with a vowel is(a) 3/14 (b) 1/14 (c) 5/14 (d) 4/11

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Q168. The probability that in a random arrangement of the letters of the word `SUNDAY’ the letters A occupies first place and Y occupies last place is(a) 11/30 (b) 5/6 (c) 5/14 (d) 1/30

Q169. Four letters of the words `THURSDAY’ are arranged in all possible ways. The probability that the word formed at random is `HURT’ is(a) 1/1680 (b) 3/1680 (c) 1/3 (d) 5/14

Q170. If the letters A, C, E, E, M, T are written at random then the probability of making EAMCET is(a) 1/120 (b) 1/180 (c) 1/360 (d) none

Q171. If the letters of the word `ARTICLE’ are arranged at random then the probability that the consonants may occupy odd places is(a) 1/30 (b) 1/35 (c) 1/42 (d) 1/21

Q172. If the letters of the word `FAILURE’ are arranged at random then the probability that the constants may occupy odd position is(a) 4/21 (b) 4/42 (c) 4/35 (d) 9/21

Q173. If the letters of the word `VECTOR’ are arranged at random then the probability that the vowels may occupy even places is(a) 1/5 (b) 4/5 (c) 1/20 (d) 19/20

Q174. The probability that in a random arrangement of letters of the word COLLEGE the two E’s and two L’s do not come together is(a) 18/21 (b) 20/21 (c) 19/21 (d) none

Q175. If the letters of the word `ARRANGE’ are arranged at random then the probability that the two R’s and two A’s come together is(a) 4/7 (b) 1/7 (c) 2/21 (d) 3/7

Q176. If the letters of the word ARRANGE are arranged at random then the probability that the two R’s come together is(a) 5/7 (b) 2/7 (c) 6/7 (d) 4/7

Q177. 4 books of mathematics and 2 of Electronics are to be arranged at random on a shelf. The probability that the books on Electronics are not together is(a) 2/3 (b) 1/3 (c) 3/4 (d) 1/4

Q178. 4 books on physics, 3 on Electronics and 2 on Chemistry are to be arranged at random on shelf. The probability that all the books of the same subject are together is(a) 1/105 (b) 2/9 (c) 1/210 (d) 3/210

Q179. Seven books on Physics and four on Mathematics are to be arranged at random on a shelf. The probability that the books on MATHEMATICS do not stands next to each other is(a) 7/33 (b) 2/11 (c) 8/33 (d) 1/33

Q180. Five Engineering four mathematics, two chemistry books are placed on a table at random. The probability that the books of each kind are all together is(a) 2!5!4!/11! (b) 3!5!4!2!/11!(c) 5!4!2!/10! (d) 3!5!4!2!/10!

Q181. Dialing a telephone numbers an old man forget the last two digits remembering only that these are different dialed at random. The probability that the number dialed correctly is(a) 1/45 (b) 1/100 (c) 1/90 (d) none

Q182. A five digit number is formed by the digits 1,2,3,4,5,6 and 8. The probability that the numbers has even digit at both ends is(a) 2/7 (b) 3/7 (c) 4/7 (d) none

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Q183. Seven papers are to be set for an examination of which three papers are of Language. The probability that in the time table the three language papers are consecutive is(a) 6/7 (b) 5/7 (c) 1/7 (d) 2/7

Q184. Seven papers are to be set for an examination of which three papers are of language. The probability that in the time table the three language papers are not consecutive is(a) 1/7 (b) 5/7 (c) 2/7 (d) 6/7

Q185. 10 papers are to be set at an examination, two of which are in Mathematics. If the papers are arranged at random. The probability that the two papers in Mathematics are not consecutive is(a) 3/7 (b) 4/5 (c) 1/5 (d) 4/7

Q186. Seven Papers are to be set for an examination of which three papers are of language. The probability that in the time table no two of the language papers are together is(a) 1/7 (b) 4/5 (c) 2/7 (d) 5/7

Q187. An organisation consist of 25 members including 4 doctors. A committee of 4 is formed at random. The probability that the committee contains exactly two doctors is(a) 126/1265 (b) 123/1265 (c) 127/1265 (d) 126/1265

Q188. In college there are 20 professors including the principle and the vice-principle from whom a committee of five is to be formed at random. The probability that the committee will contain the principle but not the vice principle is(a) 21/38 (b) 15/76 (c) 1/19 (d) 17/76

Q189. A committee of 4 boys and 3 girls in chosen from 8 boys and 5 girls. One of the boys is a brother of one of the girls. Probability that both are in the committee is(a) 0.3 (b) 0.4 (c) 0.5 (d) 0.8

Q190. If the letters of the words `REGULATION’ be arranged at random. The probability that three will be exactly four letters between R and E is there(a) 1/5 (b) 1/2 (c) 1/9 (d) 1/10

Q191. The chance that doctor A will diagnose a disease X correctly is 60%. The chance that a patient will die by his treatment after correct diagnosis is 40%, and the chance of death by wrong diagnosis is 70%. A patient of doctor A, who had disease X, died. The probability that his disease was diagnosed correctly is:(a) 16/25 (b) 20/25 (c) 6/13 (d) 6/8

Q192. Out of 40 consecutive natural numbers two are chosen at random. Probability that the sum of the numbers is odd is(a) 14/29 (b) 20/39 (c) 1/2 (d) none

Q193. Three integers are chosen at random from the first 20 integers. Then probability their product is even is(a) 2/19 (b) 3/29 (c) 17/19 (d) 4/19

Q194. Out of 30 consecutive integers 2 are chosen at random. The probability that their sum is odd is(a) 14/29 (b) 16/29 (c) 15/29 (d) 10/29

Q195. The probabilities Mr. X and Mr. Y not living for one more year are 1/ 9 and 1/7 respectively. The probability of living one more year of either one or both is(a) 2/21 (b) 6/63 (c) 15/63 (d) 62/63

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Q196. You are given a box with 20 cards in it. 10 of these cards have letters I printed on them. The order ten have the letter T printed on them If you pick up 3 cards at random and keep them in same order, the probability of making the word I.I.T. is(a) 9/80 (b) 1/8 (c) 4/27 (d) 5/38

Q197. A husband and his wife appear in an interview for two vacancies. The probability of their selection are respectively ¼ and 1/3. What is the probability that only one of them will be selected?(a) 5/11 (b) 5/12 (c) 5/13 (d) 5/14

Q198. The probability that the 13th day of a randomly chosen month is Friday is(a) 1/12 (b) 1/7 (c) 1/84 (d) none

Q199. The probability of happening or `not happening’ of an event is(a) 1 (b) 2 (c) 1/3 (d) none

Q200. The probability of a sure event is(a) 1 (b) 2 (c) 1/2 (d) unlimited

ANSWERS 4.2

1.(a) 2.(d) 3.(1/9) 4.(a) 5.(d) 6.(b) 7.(b) 8.(d) 9.(d) 10.(6/36) 11.(b) 12.(d) 13.(a) 14.(d) 15.(b) 16.(b) 17.(b) 18.(d) 19.(b) 20.(a) 21.(b) 22.(c) 23.(c) 24.(b) 25.(b) 26.(c) 27.(a) 28.(b) 29.(a) 30.(b) 31.(c) 32.(b) 33.(b) 34.(c) 35.(a) 36.(d) 37.(c) 38.(b) 39.(b) 40.(d) 41.(b) 42.(a) 43.(d) 44.(d) 45.(b) 46.(a) 47.(c) 48.(b) 49.(a) 50.(a) 51.(c) 52.(d) 53.(b) 54.(b) 55.(c) 56.(c) 57.(a) 58.(c) 59.(c) 60.(a) 61.(c) 62.(25/102) 63.(a) 64.(b) 65.(a) 66.(c) 67.(c) 68.(c) 69.(d) 70.(a) 71. (c) 72.(b) 73.(b) 74.(b) 75.(a) 76.(b) 77.(c) 78.(a) 79.(c) 80.(b) 81.(a) 82.(b) 83.(d) 84.(c) 85.(a) 86.(d) 87.(d) 88.(a) 89.(b) 90.(b) 91.(b) 92.(c) 93.(b) 94.(c) 95.(d) 96.(c) 97.(b) 98.(b) 99.(c) 100.(c) 101.(c) 102.(d) 103.(a) 104.(d) 105.(c) 106.(b) 107.(c) 108.(c) 109.(b) 110.(c) 111.(a) 112.(b) 113.(d) 114.(d) 115.(a) 116.(d) 117 (b) 118.(a) 119.(a) 120. (b) 121.(b) 122.(5/6) 123.(c) 124.(d) 125.(b) 126.(c) 127.(c) 128.(a) 129.(c) 130.(b) 131.(cancelled) 132.(d) 133.(a) 134.(d) 135.(c) 136.(b) 137.(c) 138.(b) 139.(d) 140.(b) 141.(c) 142.(b) 143.(b) 144.(a) 145.(b) 146.(b) 147.(d) 148.(c) 149.(a) 150.(c) 151.(c) 152.(b) 153.(a) 154.(c) 155.(a) 156.(c) 157.(a) 158.(d) 159.(b) 160.(b) 161.(b) 162.(c) 163.(a) 164.(c) 165.(c) 166.(a) 167.(a) 168.(d) 169.(a) 170.(c) 171.() 172.(c) 173.(a) 174.(c) 175.(c) 176.(b) 177.(a) 178.(c) 179.(a) 180.(b) 181.(c) 182.(a) 183.(c) 184.(d) 185.(b) 186.(c) 187.(d) 188.(b) 189.(c) 190.(c) 191.(c) 192.(d) 193.(a) 194.(c) 195.(d) 196.(d) 197.(b) 198.(b) 199.(a) 200.(a)

EXERCISE 4.3Based on the following information answer the questions 1 to 3

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The quality control department of a company has two Machines A and B. Machine A is a new one and passes only two percent of the defective products. Machine B is an old one and due to some flaw passes seven percent of the defective products. A product finds its way to the finished goods stores only after it is passed by both the machines

Q1. A class consists of 100 students, 25 of them are girls and 75 boys. 20 of them are rich and remaining poor. 40 of them are fair complexioned. What is the probability of selecting a fair complexioned rich girl?(a) 0.85 (b) 0.02 (c) 0.24 (d) 0.64

Q2. The present age of a person A is 35. The odds in favour of his living up to the age of 65 is 3:2. The age of another person B is 40 at present. The odds against his living up to the age 70 is 4:1. The probability that atleast one of them will be alive after 30 years is (a) 17/30 (b) 17/25 (c) 18/72 (d) 7/25

Q3. Doctors at a hospital have discovered a new disease that runs in families. A company has discovered two drugs X and Y for the disease. The hospital has agreed to try the drugs on 400 persons. The 400 persons were selected in such a way that each has an 80 percent chance of getting the disease if given neither drug. In the group, 100 persons were administered drug X, 100 persons were administered drug Y and 200 persons were administered both the drugs. Drug X reduces the probability of the disease occurrence by 35% and drug Y reduces the probability of the disease occurrence by 20%. The two drugs when taken together work independently. If randomly selected person in the group gets the disease in the future, what is the probability that he was given both the drugs? a) 0.64 b 0.416 c) 0.52 d) 0.4177 e) 0.65

Q4. In a certain country, it is found that an increase in the interest rates has a bearing on the sales of new apartment flats. There is 30 percent chance that interest rates will not rise and new apartment flat sales will increase next year. The probability that interest rates will increase next year is 40 percent. If interest rates do not rise in the following year, what is the probability that new apartment flat sales increase? a) 0.5 b) 0.6 c) 0.4 d) 0.30 e) 0.70

Q5. A group consists of 7 men and some women. The probability of selecting 2 women from them is 1/15. The number of women in the group is(a) 5 (b) 3 (c) 8 (d) 7

Based on the following information answer the questions 6 to 8.Mr. Krishna quality control manager of Gist Electric, questions the reliability of the two quality control checks in the food processor manufacturing process. One check is performed by a worker who manually checks the processors, and a second check is performed by a computer monitor. Krishna knows that 5 percent of the time the worker is apt to miss a defective processor and that 2 percent of the time the computer will malfunction and fail to detect defective processors.

Q6. If Krishna finds that the computer was malfunctioning, what is the probability that the worker will have missed a defective processor?a) 0.08 b) 0.05 c) 0.5 d) 0.001 e) 0.01

Q7. If he knows that the worker missed a defective processor, what is the probability that she will find the computer had malfunctioned? a) 0.5 b) 0.2 c) 0.4 d) 0.001 e) 0.02

Q8. What is the probability that the worker will miss a defective processor and the computer will malfunction at the same time, allowing a defective processor to the factory? a) 0.02 b) 0.002 c) 0.1 d) 0.001 e) 0.08

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Q9. In a study of the number of men and women employed at a plant, data shows that 65 percent of the employees are males, 40 percent of the employees are production workers, and the probability that an employee is a male production worker is 0.30. If a randomly selected employee turns out to be a production worker, what is the probability that the employee is a male?a) 0.35 b)0.60 c) 0.70 d) 0.25 e) 0.75

Q10. A bag contains 4 white and 3 black balls. Two draws of 2 balls are successively made, the probability of getting 2 white balls at first draw and 2 black balls at second draw when the balls drawn at first draw were replaced is:(a) 3/49 (b) 1/49 (c) 9/49 (d) 2/49

Q11. In a locality 65% can read Gujarati, 36% can read Hindi and 30% can read English. 18% can read Gujarati and Hindi. 17% can read Gujarati and English and 13% can read Hindi and English. 5% can read all the three languages. The probability that a person selected at random can read, at least one of the three language is(a) 0.78 (b) 0.80 (c) 0.88 (d) 0.55

Q12. There are two groups of students consisting of 4 boys and 2 girls, 3 boys and 1 girl. One student is selected from each groups. The probability of one boy and one girl being selected from each groups. The probability of one boy and one girl being selected is:

Q13. Turnkey Power Projects(I) Ltd., has submitted a bid for a 500 MW power plant to come up in one of the best industrial areas of AP. It has 75 percent chance of this contract to be awarded. If its competitor, Power Plant Corporation also submits the bid, the chances of TPP getting the contract are reduced to 0.30. There is a 50% chance of PPC submitting the bid for the project, what is the probability of TPP getting the contract?a) 0.2857 b) 0.525 c) 0.625 d) 0.325 e) 0.552

Q14 Event B is dependent upon event A. P(B)=0.60 and P(A and B)=0.30, then P(A/B) is a) 0.18 b) 0.30 c) 0.40 d) 0.50 e) 0.90

Q15. A bag contains 30 black and 20 red balls. If 14 balls are drawn from the bag without replacement then the probability of drawing 8 black balls and 6 red balls is approximatelya) 0.125 b) 0.242 c) 0.625 d) 0.75 e) 0.81

Q16. In a survey of 100 readers, it was found that 50 red magazine A, 30 read magazine B and 15 read both the magazines. What is the probability of finding a person in the group who reads neither magazine A nor magazine B?a) 0.15 b) 0.35 c) 0.65 d) 0.80 e) 0.85

Q17. A and B are mutually exclusive and collectively exhaustive events. Both A and B are dependent on event C.a) P(A and C) = 0.48, P(B and C) = 0.32What is the probability of event B happening if event C happens? a) 0.32 b) 0.40 c) 0.48 d) 0.60 e) 0.80

Q18. A box contains 20 light bulbs, 5 of which are known to be defective. Three light bulbs are selected at random without replacement. Which of the following is / are correct? a) Probability that all the three bulbs selected at random are

defective = 1/1140b) Probability that exactly one of three bulbs selected at random is

defective = 7/76.c) Probability that at least one of the three bulbs selected at random

is defective = 137/228d) Probability that at least two of the three bulbs selected at random

are not defective = 91/228.e) Probability that at lest one of the three bulbs selected at random

is not defective = 113/114.

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Q19. A and B toss a coin alternately. One who gets head first wins. The probabilities of their winning are respectively(a) 2/3, 1/3 (b) 1/2,1/2 (c)3/4,1/4 (d) 5/6,1/6

Q20. Alpha Industries Ltd. has 120 engineers on its rolls. Out of these 120 engineers, some are graduates and the remaining are postgraduates. The following details pertaining to the age group and qualification of the engineers have been provided by the personnel department of the company.

Age Group (years) Qualification TotalGraduate level Postgraduate level

Below 30 54 6 6030 to 40 12 18 30Above 40 24 6 30Total 90 30 120

One engineer is selected at random.Which of the following is correct?a) The probability that the engineer has a graduate level qualification

is 0.60.b) The probability that the engineer has a post-graduate level

qualification given that his age is over 40 years, 0.10.c) The probability that the engineer’s age is below 30 years, given

that he has a graduate level qualification is 0.75.d) The probability that the age of the engineer is between 30 and 40

years, and he has a graduate level qualification is 0.20.e) The probability that the age of the engineer is between 30 and 40

years given that he has postgraduate level qualification is 0.60.

Q21. Two unbiased dice are thrown. What is the probability that neither a doublet nor a total of 10 will appear?a) 2/9 b) 7/9 c) 1/9 d) 1/36 e) 5/36

Q22. A coin is tossed 12 times. What is the Probability of getting exactly 8 tails?a) 2/3 b) 1/12 c) 495/4,096 d)125/4,095 e)212/436

Q23. Two cards are drawn without replacement from a well shuffled pack of 52 cards. What is the probability that one is red queen and the other is a king of black colour?a) 1/52 b) 3/104 c) 2/663 d) 3/333 e)2/260

Q24. What is the probability to obtaining two heads in two throws of a single coin?a) 1/2 b) 1/4 c) 1/8 d) 1/16 e) 1/32

Q25. In a competitive examination, 30 candidates are to be selected. In all 600 candidates appear in a written test, and 100 will be called for interview. What is the probability of a person getting selected, if he has been called for interview?a) 1/6 b) 1/2 c) 1/20 d) 3/10 e) 5/6

Q26. What is the probability of picking an ace and a king from a deck of 52 cards?a) 4/52 b) 4/51 c) 8/663 d) 4/52 e) 8/748

Q27 One card is drawn from a deck of 52 cards. What is the probability of the card being either red or a king?a) 7/13 b) 1/26 c) 1/13 d) 1/2 e) 1/3Based on the following data answer the questions 28 and 29. Suppose a company hires both MBAs and non–MBAs for the same kind of managerial tasks. After a period of employment some of each category are promoted and some are not. Given below gives the proportion of company’s managers among the said classes.

Proportion Manager is the company

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Promotional status Academic QualificationMBA (A) Non-MBA (B) Total

Promoted (B)Not promoted (B)Total

0.420.280.70

0.180.120.30

0.600.401.00

Q28. What is the value of P(B|A)?a) 0.42 b) 0.70 c) 0.60 d) 0.40 e) 0.25

Q29. What is the value of P(A|B)a) 0.70 b) 0.60 c) 0.50 d) 0.40 e) 0.30

Q30. A bag contains 12 balls numbered from 1 to 12. If a ball is taken at random, what is the probability of having a ball with a number which is a multiple of either 2 or 3?a) 2/3 b) 1/2 c) 1/3 d) 1/6 e) 1/24

Q31. The data for the promotion status and academic qualification for another company is given in Table below:

Table Promotion of Managers in Another CompanyPromotional status Academic Qualification

MBA (A) Non-MBA (B)1 TotalPromoted (B)Not promoted (B)1Total

0.120.180.30

0.480.220.70

0.600.401.00

Which of the following statement is true regarding the data?a) ‘Education qualification’ and ‘promotional status’ are dependent eventsb) ‘Education qualification’ and ‘promotional status’ are independent eventsc) P(A\B) = 0.20d) P(B\A) = 0.50e) Both a and c above

Q32. A bag contains 3 red and 4 white balls. Two draws are made without replacement; what is the probability that both the balls are red?a) 3/7 b) 1/3 c) 1/7 d) 2/7 e) 4/7

Q33. Box one contain 1 white and 999 red balls. Box two contains 1 red 999 white balls. A ball is picked from a randomly selected box. If the ball is red, what is the probability that it came from Box one?a) 1/1,000 b) 999/1,000 c) 1/10 d) 9/10 e) 1/2

Q34. A can hit a target 3 times out of 5 trials. B can hit the target 2 times out of 5 trials: C can hit the target 3 times out of 4 trials. If all the three try simultaneously find the probability that at least 2 will hit the target.(a) 0.63 (b) 0.5 (c) 0.69 (d) 0.65

Q35. A box contains 2000 components of which 5% are defective. Second box contains 500 components of which 40% are defective. Two other boxes contain 1000 components each with 10% defective components. We select at random one of the above boxes and remove from it a random a single component. What is the probability that this component is defective?

a) 0.1625 b) 0.25 c) 0.05 d) 0.4 e) 0.1

Q36. Three candidates A, B, C are selected for the position of a general manager in a company whose chances of getting the appointment are in the proportion 4:2:3 respectively. The probability that A is selected will improve the office canteen is 0.3. The probability of B and C doing the same are respectively 0.5 and 0.8. What is the probability that the office canteen will be improved?

a) 0.51 b) 0.27 c) 0.11 d) 0.25 e) 0.5

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Q37. A box contains 10 white, 5 yellow and 10 black balls, is chosen at random from the box and it is noted that it is not one of the black balls. What is the probability that it is yellow?a) 1/4 b) 1/2 c) 2/3 d) 3/4 e) 1/3

Q38. For a 60 year old person living upto the age of 70, it is 7:5 against him and for another 70 years old person surviving upto the age of 80, it is 5:2 against him. The probability that one them will survive for 10 years more is:(a) 15/42 (b) 39/84 (c) 49/84 (d) 40/84

Q39. In a bolt factory machines A,B,C manufacture respectively 25% 35% and 40% of the total. Of their output 5,4,2 percent are defective blots. A bolt is drawn at random from the product and is found to be defective. What are the probabilities that it was manufactured by machines A, B, C?a) 25/69 b) 54/69 c) 2/5 d) 3/5 e) 1/8

Q40. Three machines in a factory produce respectively 20%, 50% and 30% of items daily. The percentage of defective items machines are respectively3, 2 and 5. An item is taken at random from the production and is found to be defective, the probability that it produced by machine A is(a) 6/31 (b) 3/31 (c) 1/31 (d) 2/31

Q41. A certain item is manufactured by three factories say 1, 2 and 3 It is known that 1 turns out twice as many items as 2, and that 2 and 3 turns out the same number if item (during a specified production period). It is also known that 2 percent of the item produced by 1 and 2 are defective, white 4 percent of those manufactured by 3 are defective. All the items produced are put into one stockpile, and then one item is chosen at random. What is the probability that this item is defective?

a) 0.25 b) 0.025 c) 0.00025 d) 0.005 e) 0.5Q42. A pair of dice is thrown twice. What is the probability of getting total

of 7 and 11?a) 1/27 b) 1/54 c) 1/18 d) 1/6 e) 1/108

Q43. An item is manufactured by three machines M1, M2, and M3. Out of the total manufactured during a specified production period, 50% are manufactured on

M1, 30% on M2 and 20% on M3. It is also known that 2% of the item produced by M1 and M2 are defective, while 3% of those manufactured by M3

are defective. All the items are put into one bin. From the bin, one item is drawn at random and is found to be defective. What is the probability that it was made on M1?a) 0.251 b) 0.625 c) 0.454 d) 0.125 e) 0.325

Q44. When a machine is set correctly, it produce 25% defective, otherwise it produce 60% defectives. From the past knowledge and experience, the manufacturer knows that the chances that the machine is set correctly or wrongly are 50:50. The machine was set and before commencement of production, one piece was inspected and found to be defective. What is

the probability of machine set up being correct?a) 0.21 b) 0.29 c) 0.27 d) 0.125 e) 0.425

Based on the following information answer the questions 45 and 46. There are four defective power supplies in a package of ten. If two power supplies are randomly selected one after another.

Q45. What is the probability of one defective and one good power supply being selected?a) 8/15 b) 2/5 c) 1/5 d) 4/5 e) 3/5

Q46. What is the probability of two defectives being selected?a) 2/15 b) 1/5 c) 1/4 d) 12/15 e) 2/5

Based on the following information answer the questions 47 and 48.

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A small town has two ambulances. Records indicate that the first ambulance is in service 60% of the time and the second one is in service 40% of the time.

Q47. What is the probability that when an ambulance is needed, one will not be available?a) 0.60 b) 0.24 c) 0.40 d) 0.15 e) 0.18

Q48. What is the probability that at least one ambulance will be available?a) 0.24 b) 0.64 c) 0.36 d) 0.76 e) 0.25

Q49. Place 100 marbles in a bag 35 blue, 45 red, and 20 yellow. P(blue) = .35 P(red) = .45 P(yellow) = 20 What is the probability of choosing either a red or a yellow marble from the bag?

a) 0.35 b) 0.65 c) 0.25 d) 0.75 e) 0.45

Q50. Suppose that the probability that I am in my office at any given movement of the typical school day is .65. Also, say that the probability that someone is looking for me in my office at any given movement of the school day is .15 What is the probability that during some particular moment, I am in my office and someone looks for me there?a) 0.0975 b) 0.375 c) 0.25 d) 0.75 e) 0.635

Q51. What is the probability that leap year selected at random will contain either 53 Thursday or 53 Friday?a) 2/7 b) 3/7 c) 4/7 d) 5/7 e) 1/7

Q52. There are 17 balls numbered from 1 to 7 in a bag. If a person selects one at random, what is the probability that the number printed on the ball will be an even number greater than 9?a) 1/17 b) 4/17 c) 5/17 d) 3/17 e) 6/17

Q53. If a pair of dice is thrown, find the probability that the sum is neither 7 nor 11?a) 7/9 b) 2/9 c) 1/3 d) 4/9 e) 5/9

Q54. Two unbiased dice are thrown. Find the probability that the sum of the faces is not less than 10.a) 1/6 b) 2/36 c) 5/36 d) 7/36 e) 1/12

Q55. Two urns contain respectively 10 white, 6 red and 9 black balls and 3 white, 7 red and 15 black balls. One ball is drawn from each urn. Find the probability that both balls are red?a) 6/25 b) 7/25 c) 42/625 d) 13/625 e) 14/125

Q56. A certain production process produces items that are 10 percent defective. Each item is inspected before being supplied to customers but the inspector incorrectly classified an item 10% of the times. Only items classified as good are supplied. If 820 items have been supplied how many of them are expected to be defective?a) 8 b) 9 c) 10 d)11 e) 12

Q57. A ox contains 4 defective and 6 good electronic calculators. Two calculators are drawn out by one without replacement, what is the probability that the two calculators so drawn are good?a) 1/3 b) 2/3 c) 1/6 d) 2/3 e) 1/2

Q58. The record of 400 examinees is given below:Score B.A. B.Sc. B.Com. Total

Below 50Between 50 and60Above 60

902010

307030

607020

18016060

Total 120 130 150 400What is the probability that he science graduate given that his score is above 60?

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a) 1/3 b) 2/3 c) 3/4 d) 1/2 e) 4/5

Q59. There are 100 students in a college class of which 36 are boys studying statistics and 13 girls are not studying statistics. If there are 55 girls in all, find the probability that a boy picked up at random is not studying statistics.a) 0.25 b) 0.20 c) 0.33 d) 22 e) 0.19

Q60. In an examination 30% of the student have failed in mathematics 20% have failed in chemistry and 10% failed in both mathematics and chemistry. A student is selected at random. What is the probability that the student has failed in mathematics, if it is known that he failed in chemistry?a) 0.20 b) 0.30 c) 0.40 d) 0.50 e) 0.60

Q61. Three horses A, B and C are in a race. A twice as likely to win as B and B is twice as likely to win as C. What are their respective probabilities of winning the horse B? a) 2/7 b) 1/7 c) 4/7 d) 3/7 e) 5/7

Q62. A, B and C are three mutually and exhaustive events. What is P(B) if 1/3P(C)=1/2P(A)=P(B)?a) 1/6 b) 1/3 c) 1/2 d) 1/4 e)1/12

Q63. The probability of Mr. A living 20 years more is 1/5 and that of Mr. B is 1/7. The probability that at least one of them will survive 20 years hence is:(a) 12/35 (b) 1/35 (c) 11/35 (d) 14/3

Based on the following information answer the questions 64 and 65Two urns contains respectively 10 white, 6 red, and 9 black balls and 3 white, 7 red and 15 black balls. One ball is drawn from each urn.

Q64. What the probability that both balls are red?a) 6/25 b) 7/25 c) 42/125 d) 42/625 e) 207/625

Q65. What the probability that both balls are of same color?a) 1/42 b) 207/625 c) 42/125 d) 9/25 e) 15/25

Q66. A certain production process produces items that are 10% defective each item is inspected before being supplied to customers but the inspector incorrectly classifies an item 10% of the times. Only items classified as good are supplied. If 820 items have been supplied how many of them are expected to be defective?a) 10 b) 6 c) 15 d) 12 e) 9

Q67. A box contains 4 defectives and 6 good electronic calculations. Two calculators are drawn out one by without replacement.What is the probability that the two calculators so drawn are good?a) 1/3 b) 1/2 c) 2/3 d) 5/9 e) 1/4

Q68. In the above question, if one of the calculators so drawn is tested and found to be good, what is the probability that the other one is also good?a) 2/9 b) 1/3 c) 5/9 d) 4/9 e) 1/9

Q69. It is known that 40% of the students in a certain college are girls and 50% of the students are above the median height. If 2/3 of the boys are above the median height what is the probability that a randomly selected student who is below the median height is a girl? a) 0.6 b) 0.4 c) 0.2 d) 0.1 e) 0.5

Q70. A and B throw alternatively wit ha pair of dice. One who first throws a total of 9 wins. What are their respective chances of winning if A starts the game?a) 9/17 b) 8/17 c) 2/17 d) 3/17 e) 5/17

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Q71. An investment consultant predicts that the odds against the price of a certain stock will go up during the next week is 2:1 and the odds in favor of the price remaining the same are 1:3. What is the probability that the price of the stock will go down during the next week?a) 1/12 b) 5/12 c) 1/4 d) 1/3 e) 2/3

Q72. A lot contains 10 items of which 3 are defective. Three items are chosen at random one after another without replacement. Find the probability that all the three are defective?a)3/10 b) 1/60 c) 1/120 d) 7/9 e) 1/5

Q73. Out of 120 tickets numbered successfully from 1 to 120 one is drawn at random. What is the probability of getting a number, which is a multiple of 5?a) 1/5 b) 4/5 c) 2/5 d) 3/5 e) 1/120

Q74. What is the probability that over a two day period the number of requests would either be 11 or 12 if at a motor garage the records of service requests along with their probabilities are as given below?

Daily Demand Probability5 0.256 0.657 0.10

a) 0.825 b) 0.625 c) 0.5 d) 0.125 e) 0.7975

Q75. Three unbiased coins are tossed. What is the probability of obtaining at least two heads?a) 3/8 b) 1/8 c) 1/2 d) 7/8 e) 5/8

ANSWERS 4.3

1.(b) 2.(b) 3.(d) 4.(a) 5.(b) 6.(b) 7.(e) 8.(d) 9.(e) 10.(d) 11.(c) 12.(b) 13.(b) 14.(d) 15.(b) 16.(b) 17.(b) 18.(e) 19.(a) 20.(e) 21.(b) 22.(c) 23.(c) 24.(b) 25.(c) 26.(c) 27.(a) 28.(c) 29.(a) 30.(a) 31.(e) 32.(c) 33.(b) 34.(a) 35.(a) 36.(a) 37.(e) 38.(c) 39.(a) 40.(a) 41.(b) 42.(b) 43.(c) 44.(b) 45.(a) 46.(a) 47.(b) 48.(d) 49.(b) 50.(a) 51.(b) 52.(b) 53.(a) 54.(a) 55.(c) 56.(c) 57.(a) 58.(d) 59.(b) 60.(d) 61.(a) 62.(a) 63.(c) 64.(d) 65.(b) 66.(a) 67.(a) 68.(c) 69.(a) 70.(b) 71.(b) 72.(c) 73.(a) 74.(e) 75.(c)

OBJECTIVE QUESTIONS (Scanner part I)

1. There are six slips in a box and numbers 1,1,2,2,3,3 are written on these slips. Two slips are taken at random form the box. The expected values of the sum of numbers on the two slips is: a) 5 b) 3 c) 4 d) 7

2. A letter is taken out at random from the word RANGE and another is taken out from the word PAGE. The probability that they are the same letters is: a) 1/20 b) 3/20 c) 3/5 d) 3/4

3. An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of same colour is:

a) b) c) d) none

4. A card is drawn from a well shuffled pack of 52 cards. Let E1 “a king or a queen is drawn” & E2: “a queen or a jack is drawn”, then:

180


a) E1 and E2 are not independentb) E1 and E2 are mutually exclusive c) E1 and E2 are independent d) None of these

5. In a non-leap year, the probability of getting 53 Sundays or 53 Tuesdays or 53 Thursdays is:

a) b) c) d)

6. If A and B are two events and P(A) = then the value

of P(A’B’) is:

a) b) c) d)

7. The probability that there is at least one error in an account statement prepared by A is 0.3 and for B and C, they are 0.4 and 0.45 respectively. A, B and C prepared 20, 10 and 40 statements respectively. The expected number of correct statements in all is: a) 32 b) 45 c) 42 d) 25

8. From a pack of cards, two are drawn, the first being replaced before the second is drawn. The chance that the first is a diamond and the second is king is:

a) b) c) d)

9. The theory of compound probability states that for any two events A and B: a) P(A B) = P(A) x P(B)b) P(A B) = P(A) x P(B/A) c) P(A B) = P(A) x P(B/A) d) P(A B) = P(A) + P(B) – P(A B)

10. The probability of getting qualified in IIT-JEE and AIEEE by a student are

respectively . The probability that the student gets qualified for

one of the these test is:

a) b) c) d)

11. Amitabh plays a game of tossing a dice. If the number less than 3 appears, he is getting Rs.a, otherwise he has to pay Rs.10. If the game is fair, find a:a) 25 b) 20 c) 22 d) 18

12. Suppose E and F are two events of a random experiment. If the probability of occurrence of E is 1/5 and the probability of occurrence of F given E is 1/10, then the probability of non-occurrence of at least one of the events E and F is:

a) b) c) d)

13. A bag contains 8 red and 5 white balls. Two successive draws of 3 balls are made without replacement. The probability that the first draw will produce 3 white balls and second 3 red balls is:

a) b) c) d)

14. A box contains 12 electric lamps of which 5 are defectives. A man selects three lamps at random. What is the expected number of defective lamps in his selection? a) 1.25 b) 2.50 c) 1.05 d) 2.03

15. Three identical dice are rolled. The probability that the same number will appear on each of them is:

181


a) 1/6 b) 1/12 c) 1/36 d) 1

16. Among the examinees in an examination 30%, 35% and 45% failed in Statistics, in Mathematics and in at least one of the subjects respectively. An examinee is selected at random. Find the probability that he failed in Mathematics only: a) 0.245 b) 0.25 c) 0.254 d) 0.55

17. An article consists of two parts A and B. The manufacturing process of each part is such that probability of defect in A is 0.08 and that B is 0.05. What is the probability that the assembled product will not have any defect? a) 0.934 b) 0.864 c) 0.85 d) 0.874

18. Daily demand for calculators is having the following probability distribution: Demand: 1 2 3 4 5 6Probability: 0.10 0.15 0.20 0.25 0.18 0.12Determine the variance of the demand. a) 2.54 b) 2.93 c) 2.22 d) 2.19

19. If 10 men, among whom are A and B, stand in a row, what is the probability that there will be exactly 3 men between A and B? a) 11/15 b) 4/15 c) 1/15 d) 2/15

20. The probability of an event can assume any value between: a) 0 and 1 b) -1 and 0 c) -1 and 1 d) None

21. The odds are 9:5 against a person who is 50 years living till he is 70 and 8:6 against a person who is 60 living till he is 80. Find the probability that at least one of them will be alive after 20 years:

a) b) c) d)

22. An urn contains 6 white and 4 black balls. 3 balls are drawn without replacement. What is the expected number of black balls that will be obtained? a) 6/5 b) 1/5 c) 7/5 d) 4/5

23. If P(A) = P and P(B) = q, then: a) P(A/B) q/p b) P(A/B) p/qc) P(A/B) p/q d) P(A/B) q/p

24. The probability that a trainee will remain with a company is 0.8. The probability that an employee earns more that Rs.20,000 per month is 0.4. The probability that an employee, who was a trainee and remained with the company or who earns more than Rs.20,000 per month is 0.9. What is the probability that an employee earns more than Rs. 20,000 per month given that he is a trainee, who stayed with the company? a) 5/8 b) 3/8 c) 1/8 d) 7/8

25. A random variable X has the following probability distribution: X : -2 3 1P(X=x) : 1/3 1/2 1/6Find E(X2) and E(2X + 5)a) 6 and 7 respectively b) 5 and 7 respectively c) 7 and 5 respectively d) 7 and 6 respectively

26. The limiting relative frequency of probability is:(a) Axiomatic (b) Classical(c) Statistical (c) Mathematical

27. If a probability density function is f(x) =

(a) (b) 0 (c) 1 (d) -

182


28. If: X : -2 3 1P(x) : 1/3 ½ 1/6Then find E(2x + 5) a) 7 b) 6 c) 9 d) 4

PROBABILITY (scanner Part II)

29. If A and B are two independent events and P(AUB) = 2/5; P (B) = 1/3. Find P (A)(a) 2/9 (b) -1/3 (c) 2/10 (d) 1/10

30. A bag contains 12 balls of which 3 are red 5 balls are drawn at random. Find the probability that in 5 balls 3 are red.(a) 3/132 (b) 5/396 (c) 1/36 (d) 1/22

31. A random variable X has the following probability distribution.X 0 1 2 3P(x) 0 2K 3K KThen, P (x<3) would be:(a) 1/6 (b) 1/3 (c) 2/3 (d) 5/6

32. P(A) = 2/3; P(B)= 3/5; P(AB) = 5/6. Find P (B/A)(a) 11/20 (b) 13/20 (c) 13/18 (d) None

33. If P(AB) = P (A) x P(B), then the events are:(a) Independent events (b) Mutually exclusive events(c) Exhaustive events (d) Mutually inclusive events

34. E (XY) is also known as:(a) E (X) + E(Y) (b) E (X) E (Y)(c) E (X) – E (Y) (d) E (X) E (Y)

35. In a pack of playing cards with two jokers probability of getting king of spade is(a) 4/13 (b) 4/52 (c) 1/52 (d) 1/54

36. Consider two events A and B not mutually exclusive, such that P(A) = 1/4, P(B) = 2/5, P(AB) = 1/2 , then P (A ) is(a) 3/7 (b) 2/10 (c) 1/10 (d) None of the above

37. If x be the sum of two numbers obtained when two die are thrown simultaneously then P(x 7) is(a) 5/12 (b) 7/12 (c) 11/15 (d) 3/8

38. E (13x + 9) = __________.(a) 13x (b) 13E(x) (c) 13E(x) + 9 (d) 9

39. A dice is thrown once. What is the mathematical expectation of the number on the dice?(a) 16/6 (b) 13/2 (c) 3.5 (d) 4.5

40. If P (A/B) = P(A), then A and B are (a) Mutually exclusive events (b) Dependent events(c) Independent events (d) Composite evens

41. A bag contains 3 white and 5 black balls and second bag contains 4 white and 2 black balls. If one ball is taken from each bag, the probability that both the balls are white is _______.(a) 1/3 b) ¼ c) ½ (d) None of these

42. The odds in favour of A solving a problem is 5 : 7 and Odds against B solving the same problem is 9 : 6. What is the probability that if both of them try, the problem will be solved?(a) 117/180 (b) 181/200 (c) 147/180 (d) 119/180

183


43. Consider Urn I : 2 white balls, 3 black balls.Urn II: 4 white balls, 6 black balls. One ball is randomly transferred from first to second Urn, then one ball is drawn from II Urn. The probability that drawn ball is white is (a) 22/65 (b) 22/46 (c) 22/55 (d) 21/45

44. If P (A B) = P (A), Find P (A B).(a) P (A). P(B) (b) P(A) + P (B) (c) 0 (d) P (B)

45. In how many ways a team of 5 can be made out of 7 Boys and 8 Girls, if 2 Girls are compulsory to form a Team.(a) 2,646 (b) 1,722 (c) 2,702 (d) 980

46. A bag contains 5 Red balls, 4 Blue Balls and ‘m’ Green Balls. If the random probability of picking two green balls is 1/7. What is the no. of green Balls (m).(a) 5 (b) 7 (c) 6 (d) None of the above

47. The probability of Girl getting scholarship is 0.6 and the same probability for Boy is 0.8. Find the probability that at least one of the categories getting scholarship.(a) 0.32 (b) 0.44 (c) 0.92 (d) None of the above

48. If 15 persons are to be seated around 2 round tables, one occupying 8 persons and another 7 persons. Find the number of ways in which they can be seated.

(a) (b) (c) 7! 8! (d) 2. 6! 7!

49. A coin is tossed 5 times, what is the probability that exactly 3 heads will occur.

(a) (b) (c) (d)

50. Two unbiased dice are thrown. The Expected value of the sum of numbers on the upper side is;(a) 3.5 (b) 7 (c) 12 (d) 6

51. One Card is drawn from pack of 52, what is the probability that it is a king or a queen?(a) 11/13 (b) 2/13 (c) 1/13 (d) None of these

52. In a packet of 500 pens, 50 are found to be defective. A pen is selected at random. Find the probability that it is non defective.(a) 8/9 (b) 7/8 (c) 9/10 (d) 2/3

53. Four married couples have gathered in a room. Two persons are selected at random amongst them, find the probability that selected persons are a gentleman and a lady but not a couple.(a) 1/7 (b) 3/7 (c) 1/8 (d) 3/8

Answers:1.(c) 2.(b) 3.(c) 4.(a) 5.(c) 6.(b) 7.(c) 8.(a) 9.(b) 10.(a) 11.(b) 12.(d) 13.(c) 14.(a) 15.(c) 16.(a) 17.(d) 18.(c) 19.(d) 20.(d) 21.(c) 22.(a) 23.(c) 24.(b) 25.(a) 26.(c) 27.(c) 28.(a) 29.(d) 30.(d) 31.(d) 32.(b) 33.(a) 34.(b) 35.(d) 36.(d) 37.(b) 38.(c) 39.(c) 40.(c) 41.(b) 42.(a) 43.(c) 44.(d) 45.(c) 46.(c) 47.(c) 48.(d) 49.(a) 50.(b) 51.(b) 52.(c) 53.(b)

Model paper Objectives Paper -1

1) Which of the following pairs of events are mutually exclusive? a) A: The students read in a school B: He studies Philosophy b) A: Raju was born in India B: He is fine Engineer c) A: Ruma is 16 years old B: She is a good singer d) A: Peter is under 15 years of age B: Peter is a voter of Kolkata

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2) If two events A and B are independent then

a) Equals to P (A/B) P (B/A) b) Equals to P (A) P (B)

c) Equals to P (A) P (B/A) d) Equals to P (B) P (A/B)

3) If two letters are taken at random from the word HOME, __________ is the probability that none of the letters would be vowels.

a) b) c) d)

4) A bag contains 15 one rupee coins, 25 two rupee coins and 10 five rupee coins, If a coin is selected at random from the bag, then the probability of not selecting a one rupee coin is

a) 0.30 b) 0.70 c) 0.25 d) 0.20

5) If a card is drawn at random from a pack of 52 cards, what is the chance of getting a spade or an ace?

a) b) c) 0.25 d) 0.20

6) If an unbiased coin is tossed once, then the two – events Head and Tails are: a) Mutually exclusive b) Exhaustive c) Equally likely d) All these (a), (b) and (c)

7) If P(A) = P(B), then a) A and B are the same events b) A and B must be same events c) A and B may be different events d) A and B are mutually exclusive events

8) If two unbiased dice are rolled together, what is the probability of getting no difference of points

a) b) c) d)

9) The probability that a card drawn at random from the pack of playing cards may be either a queen or an ace is

a) b) c) d) None of these

10) If the overall percentage of success in an exam is 60, what is the probability that out of a group of 4 students, at least one has passed?

a) 0.6525 b) 0.9744 c) 0.8704 d) 0.0256

11) A theoretical probability distribution a) Does not exist b) Exists only in theory c) Exists in real life d) Both (b) and (c)

12) Probability density function is always ___________.a) Greater than 0 b) Greater than equal to 0 c) Less than 0 d) Less than equal to 0

13) If an unbiased die is rolled once, the odds in favour of getting a point which is multiple of 3 is a) 1 : 2 b) 2 : 1 c) 1 : 3 d) 3 : 1

14) If A, B and C are mutually exclusive independent and exhaustive events then what is the probability that they occur simultaneously?a) 1 b) 0.50 c) 0 d) Any value between 0 and 1

15) It is given that a family of 2 children has a girl, what is the probability that the other child is also a girl?

a) 0.50 b) 0.75 c) d)

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16) Probability mass function is always a) 0 b) Greater than 0 c) Greater than equal to 0 d) Less than 0

17) If the probability of a horse A winning a race is and the probability of a horse B winning the

same race is _______ is the probability that one of the horses will win.

a) b) c) d)

18) If an unbiased coin is tossed twice, the probability of obtaining at least one tail is a) 0.25 b) 0.50 c) 0.75 d) 1.00

19) In a single throw with two dice the probability of getting a sum of five on the two dice is


20) A card is drawn from each of two well – shuffled packs of cards. The probability that at least one of them is an ace is


Answers:1 d 2 b 3 a 4 b 5 a 6 d 7 c 8 d 9 a 10 b11 a 12 b 13 c 14 c 15 c 16 c 17 a 18 c 19 a 20 b

Paper 2

1) A die was thrown 400 times and ‘six’ resulted 80 times then observed value of proportion is a) 0.4 b) 0.2 c) 5 d) None of these

2) What is the chance of picking a heart or a queen not of heart from a pack of 52 cards?

a) b) c) d)

3) In a single throw with two dice, chance of throwing 8 is

a) b) c) d)

4) A bag contains 10 red and 10 green balls and a ball is drawn from it. The probability that it will be green is:


5) If an event cannot take place, probability will be ________a) 1 b) - 1 c) 0 d) None of these

6) A box contains 7 red, 6 white and 4 blue balls. How many selections of three balls can be made so that none is red?a) 90 b) 120 c) 48 d) 24

7) If two events A and B are independent, the probability that they will both occur is given bya) P (A) + P (B) b) P (A) P (B) c) P (A) – P (B) d) P (A) / P (B)

8) A bag contains 30 balls numbered from 1 to 30. One ball is drawn at random. The probability that the number of the drawn balls will be multiple of 3 or 7 is

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9) A card is drawn from a pack of playing cards and then another card is drawn without the first being replaced. What is the probability of getting two hearts?


10) A pair of dice is thrown. What is the probability that the sum of the numbers obtained is more than 10?


11) Three coins are tossed. What is the probability of getting at least two tails?


12) A card is drawn from a pack of 52 cards. What is the probability that it is neither a black card nor a king?


13) A coin is tossed two times. The toss resulted in one head and one tail. What is probability that the first throw resulted in tail?


14) Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that they are both kings if the first is replaced


15) A pair of dice is thrown and sum of the numbers on the two dice comes to be 7. What is the probability that the number 3 has come on one of the dice?


16) A bag contains 5 red and 3 yellow balls. Two balls are drawn at random one after the other without replacement. The probability that both the balls drawn are yellow is


17) The chance of getting a sum of 6 in a single throw with two dice is

a) b) c) d)

18) If two letters are taken at random from the word HOME, what is the probability that none of the letters would be vowels?

a) b) c) d)

19) A card is drawn from a pack of playing cards and then another card is drawn without the first being replaced. What is the probability of getting two kings?


20) A bag contains 30 balls numbered from 1 to 30. One ball is drawn at random. The probability that the number of the drawn balls will be multiple of 5 or 7 is

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Answers:1 b 2 c 3 b 4 c 5 c 6 b 7 b 8 b 9 a 10 c11 a 12 a 13 c 14 b 15 b 16 b 17 d 18 a 19 b 20 b

Paper 3

1) If , then the two events A and B are __________. a) Mutually exclusive b) Exhaustive c) Equally likely d) Independent

2) If two events A and B are dependent, the conditional probability of B given A i.e., P (B/A) is calculated as

a) b) c) d)

3) If A and B are mutually exclusive events and P(A) = 0.3 and P(B) = 0.4, find P(A )a) 0.7 b) 0.3 c) 0.6 d) None of these

4) The probability of an event can assume any value between(a) -1 and 1 (b) 0 and 1 (c) -1 and 0 (d) None of these

5) Two cards are drawn from a well shuffled pack of playing cards. Find the probability that both are ace. (a) 1:221 (b) 2:221 (c) 10:21 (d) None of these

6) If then two event A and B are : (a) Mutually exclusive (b) Equally like (c) Independent (d) Exhaustive

7) If P (AB)=P(A) x P(B), two events A and B said to be (a) Dependent (b) Equally like (c) Independent (d) None

8) A bag contains 5 white and 10 black balls. Three balls are taken out at random. Find the probability that all three balls drawn are black.(a) 16/91 (b) 42/91 (c) 24/91 (d) None of these

9) A card is drawn from a well shuffled pack of playing Cards. Find the probability that it is a king or a Queen:(a) 1/13 (b)1/4 (c) 2/13 (d) 2/4

10) For any two events A and B (a) P(A ∩ B) < P(A) + P(B) (b) P(A ∩ B) > P(A) + P(B) (c) P(A ∩ B) ≤ P(A)+P(B) d) P(A ∩ B) ≥ P(A) + P(B)

11) If P(A) = 6/9 then the odds against the event is ________ (a) 3/9 (b) 6/3 (c) 3/6 (d) 3/15

12) A dice is tossed thrice, if getting a four is considered a success, find the variance of probability distribution of number of success.(a) ½ (b) ¼ (c) 5/12 (d) 7/12

13) A pair of dice is rolled. If the sum on the dice is 9. Find the probability that one of the dice showed 3. (a) 1/9 (b) ¼ (c) ½ (d) 1

14) Probability of throwing an even number with an ordinary six faced dice is (a) 1 (b) -½ (c) ½ (d) 0

15) P(B/A) is defined only if ____(a) A is pure event (b) B is a sure event(c) B is an impossible event (d) A is not an impossible event

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16) Candidate is selected for interview for 3 post, for the first there are 3 candidates, for the second there are 4 and for the third there are 2. What are the chances of his getting at least one. (a) ¾ (b) 2/3 (c) 1/10 (d) 1

17) If P (x/y) = p(x), then (a) x is independent of y b) y is independent of x (c) y is dependent of x (d) Both (a) and (b)

18) If a pair of dice is thrown, the probability that the sum is neither 7 nor 11 is _________(a) 7/9 (b) 5/9 (c) 11/9 (d) 2

19) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of getting a two of heart or one of diamond. (a) 1/26 (b) 2/51 (c) 2/26 (d) 1/52

20) If 15 dates are chosen at random, then the probability of getting two Friday is (a) 0.13 (b) 0.38 (c) 0.47 (d) None of these

Answers:1 a 2 c 3 a 4 d 5 a 6 a 7 c 8 c 9 c 10 a11 c 12 c 13 c 14 c 15 d 16 a 17 d 18 a 19 a 20 aPaper 41) When an event is decomposable into a number of simple events, then it is called a compound

event? (a) True (b) False (c) Both (d) None of these

2) If an unbiased coin is tossed once, then the two events head and tail are: (a) Mutually exclusive (b) Exhaustive (c) Equally likely (d) All of these

3) A bag contained 20 discs numbered in 1 to 20. A disc is drawn from the bag. The probability that the number on it is a multiple of 3 is _________ (a) 5/10 (b) 2/5 (c) 1/5 (d) 3/10

4) Out of numbers 1 to 120, one is selected at random, what is the probability that it is divisible by 8 or 10. (a) 23/120 (b) 18/125 (c) 32/120 (d) None of these

5) If A and B are two mutually exclusive events, then P(A U B) = P (A) + P (B) (a) True (b) False (c) P(A U B) = P (A/B) (d) None of these

6) Probability of occurrence of A as well as B is denoted by _________ (a) P (AB) (b) P (A+B) (c) P (A/B) (d) None of these

7) Three horses A, B and Care in a race, A is twice as likely to win as B and B is twice as likely to win as C. What is the possibility of C winning the race? (a) 1/7 (b) 3/7 (c) 2/5 (d) 2/7

8) A class consists of 10 boys and 20 girls of which half the boys and half the girls have blue eyes. Find the probability that a student chosen random is a boy and has blue eyes. (a) 1/6 (b) 3/5 (c) ½ (d) None of these

9) There are four hotels in a certain city. If 3 men check into hotels in a day, what is the probability that they each are into a different hotels. (a) 0.050 (b) 0.375 (c) 0.675 (d) 0.5225

10) A number is selected from the numbers 1,2,3,4 ……..25. The probability for it to be divisible by 4 or 7 is.(a) 3/25 (b) 9/25 (c) 1/25 (d) None of these

11) Suresh is selected for three different posts. For the First post there are 2 candidates, for the second post there are 3 candidates, for the third post there are 10 candidates. The probability, that Suresh would be selected, is (a) 0.7 (b) 0.5 (c) 0.6 (d) None of these

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12) Eight balls are distributed at random is three containers. The probability, that the first container would contain three balls, is(a) 0.37 (b) 0.17 (c) 0.27 (d) None of these

13) There are six slips in a box and numbers 1, 1, 2, 2, 3, 3 are written on these slips. Two slips are taken at random from the box. The expected value of the sum of numbers on the two slips is:

a) 5 b) 3 c) 4 d) 7

14) A letter is taken out at random from the word RANGE and another is taken out from the word PAGE. The probability that they are the same letters is:

a) 1/20 b) 3/20 c) 3/5 d) 3/4 15) An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at

random. The probability that they are of same colour is:

a) b) c) d) None

16) A card is drawn from a well shuffled pack of 52 cards. Let E1 “a king or a queen is drawn” & E2 “a queen or a jack is drawn”, then:

a) E1 and E2 are not independent b) E1 and E2 are mutually exclusive c) E1 and E2 are independent d) None of these

17) In a non – leap year, the probability of getting 53 Sundays or 53 Tuesdays or 53 Thursdays is:

a) b) c) d)

18) If A and B are two events and P (A) = , P (B) = , = , then the value of P is:

a) 1/4 b) 3/4 c) 5/8 d) 5/4

19) The probability that there is at least one error in an account statement prepared by A is 0.3 and for B and C, they are 0.4 and 0.45 respectively. A, B and C prepared 20, 10 and 40 statements respectively. The expected number of correct statements in all is:

a) 32 b) 45 c) 42 d) 25

20) From a pack of cards, two are drawn, the first being replaced before the second is drawn. The chance that the first is a diamond and the second is king is:

a) b) c) d)

Answers:1 a 2 d 3 d 4 a 5 a 6 a 7 a 8 a 9 b 10 b11 a 12 c 13 c 14 b 15 c 16 a 17 c 18 b 19 c 20 a

JUNE (2016)

1. If P(A) = , P (B) = , P (A B) = then P (A/B) is

(a) (b) (c) (d)

2. Two dice are tossed what is the probability that the total is divisible by 3 or 4.


3. If 2 dice are rolled simultaneously then the probability that their sum is neither 3 nor 6 is(a) 0.5 (b) 0.75 (c) 0.25 (d 0.80

4. In a game, cards are thoroughly shuffled and distributed equally among four players. What is the probability that a specific player gets all the four kings?

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(a) (b) (c) (d)

5. A bag contains 4 red and 5 black balls. Another bag contains 5 red, 3 black balls. If one ball is drawn at random from each bag. Then the probability that one red and one black ball drawn is _________.

(a) (b) (c) (d)

6. In a discrete random variable follows uniform distribution assumes only the values 8,9,11,15,18,20. Then P (X 15) is ________(a) 1/2 (b) 1/3 (c) 2/3 (d) 2/7

Answers 1. (a), 2. (a), 3. (d), 4. (b). 5. (c), 6. (c)

DEC- 2016

1. A bag contains 6 green and 5 red balls. One ball is drawn at random. The probability of getting a red ball is?


2. If P (A) = , P (B) = , P (A B) = , then find P (A B)

(a) (b) (c) (d)

3. If P (A) = , P (B) = , P(A B) = , then the events A & B are _______

(a) Independent and mutually exclusive(b) Independent but not mutually exclusive(c) Mutually exclusive but not independent(d) Neither independent nor exclusive

Answers:1.(a) 2.(b) 3.(b)

JUNE-2017

1. The probability of getting atleast one 6 from 3 throws of a perfect die is

(a) (b) (c) 1- (d)

2. For any two events A and B(a) P (A – B) = P (A) – P (B) (b) P (A – B) = P (A) – P (A B)(c) P (A – B) = P (B) – P (A B) (d) P (B – A) = P(B) + P (A B)

3. If P (A) = P (A B) = then

(a) 1/8 (b) 7/8 (c) 8/7 (d) None

Answers : 1.(D) 2.(B) 3.(A)

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