1 CONDITIONAL PROBABILITY
Lecture 4
ORIE3500/5500 Summer2011 Li
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
)
.
(1)
From the deﬁnition it is obvious why we make the assumption that
P
(
B
)
>
0
(since it is in the denominator).
Example 1: Suppose we toss a fair dice twice. What is the probability
that the sum of the 2 dice is 8? Suppose I tell you that the ﬁrst die landed
on a 3; what is the probability that the sum of the 2 dice is 8?
Ans: Let
A
be the event that the sum of the 2 dice is 8 and
B
be the event that
the ﬁrst die landed on a 3. Then
P
(
A
) = 5
/
36 and
P
(
A

B
) = (1
/
36)
/
(1
/
6)
.
Think: how to interpret
P
(
A

B
)?
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
A
, 0
≤
P
(
A

B
).
2. (Normalization)
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 Spring '11
 SONG
 Conditional Probability, Probability, Probability theory, 1%

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