Probability

Probability: The extent to which an event is likely to occur.

• p= Favorable outcomes / Total outcomes

• Probability of heads in a coin flip

– p=1/2=0.5

Probability:

• Always positive

• Always between ‘0’ and’1’

• Probability of an event happening and not happening is always equal to ‘one’.

– p + q = 1

Probability in compound events

Mutually exclusive events:

• Events that can’t happen at the same time

• Probability of event A or B

• P (A or B)=P(A) + P(B)

A dice is rolled once. What is the probability of rolling a ‘1’ or ‘2’?

p= Favorable outcomes / Total outcomesFor rolling 1

1 /6

p= Favorable outcomes / Total outcomesFor rolling 2

1 /6

Probability of rolling a ‘1’ or ‘2’?

• (1/6) + (1/6) = 1/3

What is the probability of getting a sum of ‘7’ or ‘12’ when a pair of dice is thrown once?

(6/36) + (1/36) = 7/36 = 0.194

What is the probability of a person selected from the group described in the following table being married or single?

Marital Status Women Men Total

Married 225 225 450

Single 60 60 120

Divorced 200 200 400

Widowed 15 15 30

Total 500 500 1000

A person can not be single or married at the same time. The events are mutually exclusive.

P(A)=450/1000=0.450 , P(B)=120/1000=0.120

P (A or B) = P(A) + P(B)

=0.450 + 0.120 = 0.570

Marital Status Women Men Total

Married 225 225 450

Single 60 60 120

Divorced 200 200 400

Widowed 15 15 30

Total 500 500 1000

Non-mutually exclusive events

• Events that can happen at once

P ( A or B) = P(A) + P(B) – P(A and B)

P(A and B) Probability of two favorable events happening at once

What is the probability of picking a black card or a ‘Two’ from a deck of 52 cards?

ABlack BA two

P ( A or B) = P(A) + P(B) – P(A and B)

26 black cards in a deck of 52P(A)=26/52

4 cards with the value of two in a deck of 52 P(B)=4/52

(26/52) + (4/52) – (2/52)=28/52

=0.5385

What is the probability of a person selected from the group described in the following table being married or a man?

Marital Status Women Men Total

Married 225 225 450

Single 60 60 120

Divorced 200 200 400

Widowed 15 15 30

Total 500 500 1000

A person can be both married and also a man. So the events are not mutually exclusive

P(Married or Man) = P(Married) + P(Man) – P(Married and Man)

=0.50 + 0.45 – 0.225

=0.725

Marital Status Women Men Total

Married 225 225 450

Single 60 60 120

Divorced 200 200 400

Widowed 15 15 30

Total 500 500 1000

Independent events

• The occurrence of one event does not affect the occurrence of the others

P(A or B) = P(A) x P(B)

What is the probability of getting two heads in a row if a coin is flipped twice?

P(1.Heads ve 2. Heads)=P(1. Heads) x P(2. Heads)

=(1/2)x(1/2)=1/4

=0.25

For three coin flips ½ x ½ x ½ =1/6

What is the probability of randomly selecting a single man from the group described in the table below?

Marital Status Women Men Total

Married 225 225 450

Single 60 60 120

Divorced 200 200 400

Widowed 15 15 30

Total 500 500 1000

P(single)=120/1000=0.120 , P(man)=500/1000=0.500

P(single and man) = P(single) x P(man)

=0.120 x 0.500

=0.060

Marital Status Women Men Total

Married 225 225 450

Single 60 60 120

Divorced 200 200 400

Widowed 15 15 30

Total 500 500 1000

Dependent events

• The outcome of the first event affects the outcome for the second event.

P(A and B)= P(A) x P(B│A)

P(B│A)Probability of the favorable outcome for B after the outcome of A

Two cards have been drawn from the deck of 52 cards without replacing the first one back. What is the probability of drawing two ‘One’s?

P(A and B)= P(A) x P(B│A)

=(4/52) x (3/51)

=0.0045

Two cards have been drawn from the deck of 52 cards without replacing the first one back. What is the probability of getting first card as king and second card as queen?

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Clearly, the two events are dependent.

Let A be the event of drawing the king first, so P(A)=4/52

Now one card is drawn already so we are left with 51 cards only.

Let B be the event of drawing a queen next, so P(B) = 4/51

P(A and B)= P(A) x P(B│A)= 4/52 x 4/51=16/2652=0.006

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In a certain test 5 out of 20 students scored an ‘A’. We chose three students at random out of the 20 students without replacement. Find the probability that all three are the ones who scored an ‘A’.

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It is clear that all three events are dependent events.

Let A be the event of choosing first student with grade ‘A′, so P(A) = 5/20

Now number of students is equal to 19 and number of students with grade ‘A’ are 4.

Let B be the event of choosing second student with grade ‘A′, so P(B)= 4/19

.

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Now number of students is equal to 18 and number of students with grade ‘A’ are 3.

Let C be the event of choosing third student with grade ‘A′, so P(C)= 3/18

Hence the compound probability of all three is given by:

P(A and B and C) = P(A) X P(B│A) P((C)│(B ∩ A)) =5/20 x 4/19 x 3/18=1/114= 0.0087

Hence the probability of choosing all three students with grade ‘A’ is 0.0087

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Conditional Probabilities

Suppose that in the general population, there are 51% men and 49% women, and that the proportions of colorblind men and women are shown in the probability table below:

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If a person is drawn at random from this population and is found to be a man (event B), what is the probability that the man is colorblind (event A)?

The probability of being colorblind, given that the person is male, is 4% of the 51%.

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What is the probability of being colorblind, given that the person is female?

Now we are restricted to only the 49% of the population that is female

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A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement, what is the probability that all four walk to school?

(7/30)x(6/29)x(5/28)x(4/27)=1/783

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A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement, what is the probability that the first two chosen walk to school and the next two do not walk to school?

28

00323.27

22

28

23

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6

30

7

The Difference between Mutually Exclusive

and Independent Events

• When two events are mutually exclusive, they cannot both happen together when the experiment is performed.

• Once the event B has occurred, event A cannot occur, so that P(A I B)= 0, or vice versa.

• The occurrence of event B certainly affects the probability that event A can occur

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• Therefore, mutually exclusive events must be dependent.

• When two events are mutually exclusive,

– P(A ∩ B) = 0 and P(A U B) = P(A) + P(B).

• When two events are independent,

– P(A∩B)=P(A)P(B), and P(A U B)= P(A) + P(B) - P(A)P(B).

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Mutually Exclusive Events

• P (A or B)=P(A) + P(B)

Non-Mutually Exclusive Events

• P ( A or B) = P(A) + P(B) – P(A ve B)

Independent Events

• P(A or B) = P(A) x P(B)

Dependent Events

• P(A and B)= P(A) x P(B│A)