ProbabilityChapter 11

1

Conditional Probability

Section 11.4

2

Conditional Probability If A and B are events in a sample space U

and P(B) ≠ 0, the conditional probability of A, given B, is denoted by P(A|B) and is defined by

)(

)()|(

BP

BAPBAP

3

ExamplesA family has 3 children. If each of the outcomes isequally likely, find the probability that the family has 3 girls given that

a. the first child is a girl.b. the first child is a boy.c. the first and second child are girls.

4

1

2

1

2

11)girl1|girls 3( .a st P

02

1

2

10)boy1|girls 3( .b st P

2

1

2

111)girls2 and1|girls 3( .c ndst P

4

ExamplesA recent survey of 400 instructors at a major university revealed the data shown in the following table. Based on the data, what are the probabilities of the following?

a. An instructor received a good evaluation, given the instructor was tenured.

b. An instructor received a poor evaluation, given the instructor was tenured.

5

Solutions

10

7

240

168

16872

168)tenured|poor( .b

P

10

3

240

72

16872

72)tenured|good( .a

P

StatusGood

EvaluationsPoor

Evaluations

Tenured 72 168

Nontenured 84 76

6

Example: Companies A, B, and C produce 29%, 24%,

and 47% respectively of the major appliances sold in a certain area. In that area, 1.90% of the Company A appliances, 2.30% of the Company B appliances, and 1.20% of the Company C need service within the first year. Suppose an appliance that needs service within the first year is chosen at random, find the probability it was manufactured by Company A.

7

Solution:Co. A

service

Co. B

service

Co. C

service

0.29

0.24

0.0120

0.0190

0.0230

0.47

0.00551

8

0.00552

0.00564

3305.0)00564.000552.000551.0(

00551.0)service needs|A Co.(

P

Example:

The student body at Overpower Tech is 46% male. Although 37% of the men major in business, only 21% of the women select that major. Find the probability that a non business major is a man.

9

Solution:men

business

not business

women

business

not business

10

0.46

0.79

0.37

0.210.54

0.63

0.1702

4045.0)4266.02898.0(

2898.0)businessnot |man(

P

0.2898

0.1134

0.4266

Notice that the branches always add to give a sum of one.

Examples Two dice were thrown, and a friend tells us

that the numbers that came up were different. Find the probability that the sum of the numbers was

a. 4. b. 7. c. an even number. d. an odd number.

11

12

Solutions

15

1

30

2)different|4of sum( .a P

13

5

1

30

6)different|7of sum( .b P

5

2

30

12)different|sumeven ( .c P

5

3

30

18)different|sum odd( .d P

END