Download Conditional Probability

Conditional Probability
Test 1
Conditional Probability
Test 1 Results
You will get back your test 1 papers on Friday.
There is a generous nonlinear curve of the scores: if x is your raw score,
your grade out of 100 can be computed as
r
r
100x
x
= 50
x̃ = 10
44
11
1
2
3
4
5
6
7
8
9
10
11
|
|
|
|
|
|
|
|
|
|
|
15
21
26
30
34
37
40
43
45
48
50
Arthur Berg
12
13
14
15
16
17
18
19
20
21
22
|
|
|
|
|
|
|
|
|
|
|
52
54
56
58
60
62
64
66
67
69
71
23
24
25
26
27
28
29
30
31
32
33
|
|
|
|
|
|
|
|
|
|
|
Conditional Probability
72
74
75
77
78
80
81
83
84
85
87
34
35
36
37
38
39
40
41
42
43
44
|
|
|
|
|
|
|
|
|
|
|
88
89
90
92
93
94
95
97
98
99
100
2/ 11
Test 1
Conditional Probability
Grade Distributions, Old and New
Arthur Berg
Conditional Probability
3/ 11
Test 1
Conditional Probability
Unemployment Rate
9.4 × 106 (job-seekers)
= 6.1%
154 × 106 (job-seekers + employed)
Arthur Berg
Conditional Probability
4/ 11
Test 1
Conditional Probability
Unemployment by Education
Arthur Berg
Conditional Probability
5/ 11
Test 1
Conditional Probability
Relative Frequencies of Employment by Education
Arthur Berg
Conditional Probability
6/ 11
Test 1
Conditional Probability
Example 3.1
Conditional distribution on Men and conditional distribution on Women:
Arthur Berg
Conditional Probability
7/ 11
Test 1
Conditional Probability
Probability of Two Girls
1
What is the probability that a family with two children has two girls?
2
If you know the given family has at least one girl, what is the probability of
the family having two girls?
Definition
If A and B are any two events, then the conditional probability of A given B,
denoted by P(A|B), is
P(AB)
P(A|B) =
P(B)
provided that P(B) > 0.
P(AB) can then be computed in two ways as:
P(AB) = P(A|B)P(B)
P(AB) = P(B|A)P(A)
Arthur Berg
Conditional Probability
8/ 11
Test 1
Conditional Probability
Probability of Two Girls
1
2
What is the probability that a family with two children has two girls?
S = {BB, BG, GB, GG} where each is equally likely, so the probability is
1/4.
If you know the given family has at least one girl, what is the probability of
the family having two girls?
Definition
If A and B are any two events, then the conditional probability of A given B,
denoted by P(A|B), is
P(AB)
P(A|B) =
P(B)
provided that P(B) > 0.
P(AB) can then be computed in two ways as:
P(AB) = P(A|B)P(B)
P(AB) = P(B|A)P(A)
Arthur Berg
Conditional Probability
8/ 11
Test 1
Conditional Probability
Probability of Two Girls
1
2
What is the probability that a family with two children has two girls?
S = {BB, BG, GB, GG} where each is equally likely, so the probability is
1/4.
If you know the given family has at least one girl, what is the probability of
the family having two girls?
Definition
If A and B are any two events, then the conditional probability of A given B,
denoted by P(A|B), is
P(AB)
P(A|B) =
P(B)
provided that P(B) > 0.
P(AB) can then be computed in two ways as:
P(AB) = P(A|B)P(B)
P(AB) = P(B|A)P(A)
Arthur Berg
Conditional Probability
8/ 11
Test 1
Conditional Probability
Probability of Two Girls
1
2
What is the probability that a family with two children has two girls?
S = {BB, BG, GB, GG} where each is equally likely, so the probability is
1/4.
If you know the given family has at least one girl, what is the probability of
the family having two girls? S = {BG, GB, GG} where each is equally
likely, so the probability is 1/3.
Definition
If A and B are any two events, then the conditional probability of A given B,
denoted by P(A|B), is
P(AB)
P(A|B) =
P(B)
provided that P(B) > 0.
P(AB) can then be computed in two ways as:
P(AB) = P(A|B)P(B)
P(AB) = P(B|A)P(A)
Arthur Berg
Conditional Probability
8/ 11
Test 1
Conditional Probability
Probability of Two Girls
1
2
What is the probability that a family with two children has two girls?
S = {BB, BG, GB, GG} where each is equally likely, so the probability is
1/4.
If you know the given family has at least one girl, what is the probability of
the family having two girls? S = {BG, GB, GG} where each is equally
likely, so the probability is 1/3.
Definition
If A and B are any two events, then the conditional probability of A given B,
denoted by P(A|B), is
P(AB)
P(A|B) =
P(B)
provided that P(B) > 0.
P(AB) can then be computed in two ways as:
P(AB) = P(A|B)P(B)
P(AB) = P(B|A)P(A)
Arthur Berg
Conditional Probability
8/ 11
Test 1
Conditional Probability
Example 3.2
Example (Example 3.2, p. 64)
There are four batteries, and one is defective. Two are to be selected at random
for use on a particular day. Find the probability that the second battery selected
is not defective, given that the first was not defective.
Let Ni denote the event that the ith battery selected is nondefective.
P(N2 |N1 ) =
P(N1 N2 )
P(N1 )
P(N1 N2 ) =
3·2
1
=
4·3
2
where
and
P(N1 ) =
3
4
Therefore P(N2 |N1 ) = 32 .
Arthur Berg
Conditional Probability
9/ 11
Test 1
Conditional Probability
Example 3.2
Arthur Berg
Conditional Probability
10/ 11
Test 1
Conditional Probability
Three axioms of probability
Conditional probabilities satisfy the three axioms of probability.
(i) 0 ≤ P(A|B) ≤ 1
(ii) P(S|B) = 1
(iii) If A1 , A2 , . . . are mutually exclusive events, then so are A1 B, A2 B, . . . , and
! X
∞
∞
[
P
Ai B =
P(Ai |B)
i=1
Arthur Berg
i=1
Conditional Probability
11/ 11