1 CONDITIONAL PROBABILITY
Lecture 3
ORIE3500/5500 Summer2009 Chen
Conditional Probability and Bayes’ Rule
1 Conditional Probability
If
A
and
B
are two events such that
P
(
B
)
>
0, then the conditional proba
bility of
A
given
B
,
P
(
A

B
) is deﬁned as
P
(
A

B
) =
P
(
A
∩
B
)
P
(
B
)
.
From the deﬁnition it is obvious why we make the assumption that
P
(
B
)
>
0(since it is in the denominator). If
P
(
B
) = 0 then conditional probability
of
A
given
B
is not deﬁned and for computations you can take it to be any
number. But any how note that
P
(
A
∩
B
) =
P
(
B
)
P
(
A

B
)
(1)
always holds. If
P
(
B
)
>
0 then it follows from the deﬁnition of conditional
probability and if
P
(
B
) = 0 then both sides of the equation equals 0. When
this is generalized to higher number of events the phenomenon is often called
the chain rule
.
Theorem
(Chain Rule)
.
If
A
1
,A
2
,...,A
n
are events, then
P
(
A
1
∩ ··· ∩
A
n
) =
P
(
A
1
)
P
(
A
2

A
1
)
P
(
A
3

A
1
∩
A
3
)
···
P
(
A
n

A
1
∩ ···
A
n

1
)
.
It is not diﬃcult to see how we arrive at this result. It follows by repeated
application of (1).
P
(
A
1
∩ ··· ∩
A
n
) =
P
(
A
1
∩ ···
A
n

1
)
P
(
A
n

A
1
∩ ···
A
n

1
)
=
P
(
A
1
∩ ···
A
n

2
)
P
(
A
n

1

A
1
∩ ···
A
n

2
)
·
P
(
A
n

A
1
∩ ···
A
n

1
)
.
.
.
=
P
(
A
1
)
P
(
A
2

A
1
)
P
(
A
3

A
1
∩
A
3
)
···
P
(
A
n

A
1
∩ ···
A
n

1
)
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document1 CONDITIONAL PROBABILITY
Conditional Probabilities satisfy the probability law
Conditional probabilities are very much similar to normal probabilities. It is
like you restrict your attention to the event
B
and scale up the probabilities
of the subsets of
B
so that it becomes a probability itself when you consider
B
to be your sample space.
1. (Nonnegativity) For any
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 WEBER
 Conditional Probability, Probability theory

Click to edit the document details