AMS 311, Lecture 7, Spring 2010
Chapter Three
Conditional Probability and Independence
3.2
Conditional Probabilities
Definition
: If
,
0
)
(
F
P
the conditional probability of
E
given
F
, denoted by
)

(
F
E
P
,
is
.
)
(
)
(
)

(
F
P
EF
P
F
E
P
=
This definition satisfies the axioms of probability theory.
Example 1. From the set of all families with two children, a family is selected at random
and found to have a girl. What is the probability that the other child of the family is a
girl? Assume that in a twochild family all sex distributions are equally probable.
Answer: 1/3.
Example 2. From the set of all families with two children, a child is selected at random
and is found to be a girl. What is the probability that the second child of this girl’s family
is also a girl? Assume that in a twochild family all sex distributions are equally probably.
Answer: ½.
Law of Multiplication
:
).

(
)
(
)
(
F
E
P
F
P
EF
P
=
The (General) Multiplication Rule
:
If
,
0
)
(
2
1
N
E
E
E
P
then
).

(
)

(
)

(
)
(
)
(
1
2
1
1
2
3
1
2
1
1
3
2
1


=
n
n
n
n
E
E
E
E
P
E
E
E
P
E
E
P
E
P
E
E
E
E
E
P
3.3.
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 Spring '08
 Tucker,A
 Conditional Probability, Probability, Probability theory

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