Probability

Probability It allows us to quantify the

variability in the outcome of an experiment whose exact result can’t be predicted with certainty.

Definitions

Random Experiment

A random experiment or trial is one which when conducted successively under the identical conditions, the result is not unique but may be any one of the various possible outcomes.

Example: Tossing a fair coin in an experiment.

Sample space: The set of all possible outcomes is a

sample space.

Event:Outcome or combination of

outcomes

Outcome:The result of an event which we

finally achieved is called an outcome or a sample point.

Types of Events

Mutually exclusive events:Two events are said to be mutually exclusive or incompatible, when both cannot happen simultaneously in a single trial. AB=

Example: Toss of a coin (either head will occur or tail in a single throw)

Independent and dependent events Two or more events are independent

when the outcome of one does not affect, and is not affected by the other.

Example: if a coin is tossed twice, the result of the second throw would not be affected by the result of the first throw

P(AB)=P(A).P(B) • Dependent events: occurrence of one

event affects probability of happening of other.

Equally likely events: events are called equally likely if they have the same chance of occurrence.

Example: Throw of unbiased coin (both head and tail have equal chance of occurrence.)

Probability Numerical measure (between 0 and 1

inclusively) of the likelihood or chance of occurrence of an uncertain event

P(E) = (NO. OF FAVOURABLE OUTCOMES) (NO. OF TOTAL OUTCOMES)

0 P(E) 1

Questions for practice A uniform die is thrown. Find the probability that

the number on it is (i) Five (ii) greater than 4 (iii)Even no.

2. In a single throw with two uniform dice, find the probability of throwing

(i) Both the dice show the same number (ii)A total of Eight (iii) a total of 13 (iv) Total of the numbers on the dice is any number

from 2 to 12, both inclusive.

3. A bag contains 4 white, 5 red and 6 green balls.Three balls are drawn at random.What is the chance that a white, a red and a green ball is drawn?

4. Four cards are drawn at random from a pack of 52 cards.Find the probability that

They are a king, a queen , a jack or an ace.

Two are kings and two are aces. All are diamonds. Two are red and two are black. There are two cards of clubs and

two cards of diamonds.

5. Three unbiased coins are tossed.What is the probability of obtaining:

All heads Two heads One head At least one head At least two heads All tails

6. Five men in a company of 20 are graduates.If 3 men are picked out of the 20 are random, what is the probability that they all are graduates?What is the probability of at least one graduate?

7. Three groups of workers contain 3 men and one women, 2 men and 2 women, and 1 man and 3 women respectively.One worker is selected at random from each group.What is the probability that the group selected consists of 1 man and 2 women?

Answers1. 1/6, 1/3, ½2. 1/6, 5/36, 0,13. 24/91

4. 256/52C4, 4C2 x 4C2 / 52C4 , 13C4/ 52C4, 26C2 x 26C2 / 52C4, 13C2 x 13C2 / 52C4.5. 1/8, 3/8, 3/8,7/8, ½, 1/86. 1/114, 137/2287. 13/32

Some Important Results-

• 0 P(A) 1 for all A• P(S) = 1

• P(Ac) = 1 – P(A) for all A• P(A B) = P(A) + P(B) – P(A B)

for all A, B

Addition theorem: For two disjoint or mutually exclusive events A&B (i.e P(A B) =0 Since A B= )

P(A B) = P(A) + P(B) ORP(A OR B)=P(A)+P(B) OR P(A+B)=P(A)+P(B)

Theorems Of Probability

When events are not mutually exclusive i.e P(A B)≠0

P(A B) = P(A) + P(B) – P(A B) ORP(A OR B) = P(A) + P(B) – P(A AND B) OR P(A + B) = P(A) + P(B) – P(A.B)

For three events A,B & C,

P(A B C) = P(A)+P(B)+P( C )-P(A B)-P(B C)-P(A C)

+P(A B C)

Questions for practice1. From 25 tickets marked with first 25 numerals,

one is drawn at random.Find the chance that (i) Multiple of 5 or 7. (ii) Multiple of 3 or 7.

2. Of 1000 assembled components,10 has a working defect and 20 have a structural defect.There is a good reason to assume that no components has both defects.What is the probability that randomly chosen component will have either type of defect?

3. The probability that a contractor will get a plumbing contract is 2/3 and the probability that he will not get an electrical contract is 5/9.If the probability of getting at least one contract is 4/5,what is the probability that he will get both?

4. A card is drawn from a pack of 52 cards. Find the prob. of getting a king or a heart or a red card?

5. Two dice are tossed.find the probability of getting an ‘even number on first die or a total of 8’.

6. The prob. That a student passes a Physics test is 2/3 and the probability that he passes both a Physics and English test is 14/45.The probability that he passes at least one test is 4/5.What is the probability he passes the English test?

Answers1. 8/25,2/5

2. 0.03

3. 0.31

4. 7/13

5. 5/9

6. 4/9

Multiplication Theorem

If two Independent events occur simultaneously

P(A.B) = P(A).P(B) OR P(A B)=P(A).P(B) OR P(A AND B)=P(A).P(B)

If A & B are Dependent events P(A B)= P(B).P(A/B) ; P(B)>0 P(A B)= P(A).P(B/A) ; P(A)>0

P(A/B):Conditional Prob. of event ‘A’ given that B has already occurred.

More results

For two Independent events A&B P(A B)= P(A).P(B)

For three Independent events A,B&C P(A B)= P(A).P(B) P(A C)= P(A).P(C) P(B C)= P(B).P(C) P(A B C)= P(A).P(B).P(C)

For two mutually exclusive events A&B, A B= AND P(A B) =0

Results contd..

Probability of the complementary event Ac of A is given by

P(Ac)=1-P(A)

Demorgan’s Law

(AB)c = Ac Bc

(AB)c = Ac Bc

Questions for practice1. An MBA applies for job in two firms X and

Y.The probability of his being selected in firm X is 0.7 and being rejected at Y is 0.5. What is the probability that he will selected in one of the firms?

2. Probability that A can solve a problem is 4/5, B can solve it is 2/3 and C can solve

it is 3/7.If all of them try independently, find the probability that the problem is solved.

3. A bag contains 5 White and 3 Black Balls.Two balls are drawn at random one after the other without replacement.Find the probability that both balls drawn are black.

4. Find the probability of drawing a queen,a king and a jack in that order from a pack of cards in three consecutive draws,the card drawn is not been replaced?

5. The odds against manager X settling the wage dispute with the workers are 8:6 and odds in favour of Manager Y settling the same dispute are 14:16.

(i) What is the chance that neither settles the dispute, if they both try,independently of each other?

(ii) What is the probability that dispute will be settled?

6. A box contains 3 red and 7 white balls.one ball is drawn at random and in its place a ball of other color is put in the box.Now one ball is drawn at random from the box.Find the probability that it is red.

7. A husband and wife appear in an interview for two vacancies in the same post. The probability of husband’s

selection is 1/7 and that of wife’s selection is 1/5. What is the probability that(i) both will be selected

ii) only one will be selectediii)none be selected

Answers1. 0.5

2. 101/1105

3. 3/28

4. .00048

5. 32/105,73/105

6. 0.34

7. .029,0.286,0.686