J. Chen, Handout 3, STAT550, 2010
1
STAT550 Applied Probability
http://wwwrohan.sdsu.edu/ jchenyp/STAT5502010.htm
1.5 Conditional Probability
We want to ﬁnd the conditional probability of an event
A
given that some other
event
B
has occurred. The information that an event
B
has occurred may aﬀect the
probability of event
A
.
The symbol
P
(
A

B
) read ”the probability of
A
given
B
”  is used to denote a
conditional probability.
P
(
A

B
) refers to the PROBABILITY that
A
will occur given
that
B
has already occurred.
EX1.
Consider rolling a die. If we know that the outcome of a roll of a die is even,
what is the probability of the outcome 1, 6?
We deﬁne
A
is ”1, 6 appear” and
B
is ”even number appear ”. Clearly,
P
(
A
) =
2
6
,
P
(
B
) =
3
6
,
P
(
A
∩
B
) = 1
/
6 and
P
(
A

B
) =
1
3
=
1
/
6
3
/
6
=
P
(
A
∩
B
)
P
(
B
)
Therefore the conditional probability of
A
given
B
is the ratio of
P
(
A
∩
B
) and
P
(
B
).
Deﬁnition 1.51.
Let
A
and
B
be any two events deﬁned on a sample space
S
such
that
P
(
B
)
>
0. Then the conditional probability of
A
given
B
is given by
P
(
A

B
) =
P
(
A
∩
B
)
P
(
B
)
.
The deﬁnition of
P
(
A

B
) yields
P
(
A
∩
B
) =
P
(
A

B
)
P
(
B
)
Call Multiplication Rule. Also we have
P
(
A
∩
B
) =
P
(
B

A
)
P
(
A
)
Both rules allow us to use the conditional probability to evaluate the intersection
probability.
Notice that the following properties will be useful.
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2
1) Based on the deﬁnition of the conditional probability, it follows that
P
(
A

B
C
) =
P
(
A
∩
B
C
)
P
(
B
C
)
,
P
(
A

B
∩
C
) =
P
(
A
∩
B
∩
C
)
P
(
B
∩
C
)
,
P
(
A

B
∪
C
) =
P
(
A
∩
(
B
∪
C
))
P
(
B
∪
C
)
.
2) Specially, when
B
is the sample space, then
P
(
A

S
) =
P
(
A
∩
S
)
P
(
S
)
=
P
(
A
)
So the probability of
A
is a special conditional probability.
3) The conditional probability
P
(
A

B
) satisﬁes the three Axiom properties, i.e. i)
0
< P
(
A

B
)
<
1; ii)
P
(
B

B
) = 1, and iii)
P
(
A
1
∪
A
2

B
) =
P
(
A
1

B
)+
P
(
A
2

B
),
if
A
1
and
A
2
are disjoint.
Similarly, the result holds for higherorder intersections. Consider three events
A
,
B
, and
C
. The conditional probability
P
(
A
∩
B
∩
C
) can be written as
P
(
A
∩
B
∩
C
) =
P
(
A
)
P
(
B

A
)
P
(
C

A
∩
B
)
=
P
(
B
)
P
(
A

B
)
P
(
C

A
∩
B
)
=
P
(
C
)
P
(
A

C
)
P
(
B

A
∩
C
)
In general, for
n
events
A
1
,A
2
,
···
,A
n
, we have the the following formula
P
(
A
1
∩
A
2
∩ ··· ∩
A
n
)
=
P
(
A
1
)
P
(
A
2

A
1
)
P
(
A
3

A
1
∩
A
2
)
···
P
(
A
n

1

A
1
∩
A
2
∩ ···
A
n

2
)
×
P
(
A
n

A
1
∩
A
2
∩ ···
A
n

1
)
EX2.
A card is drawn from a poker deck. What is the probability that the card is a
club, given that the card is a king?
Solution:
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 Spring '08
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 Conditional Probability, Probability, Probability theory, J. Chen

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