probability chapter 11 1. conditional probability section 11.4 2
TRANSCRIPT
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.
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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