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Chapter 2, Section 7A
Conditional Probability
©
John J Currano, 01/12/2010
2
Game:
Toss a die and observe the top face
Sample Space:
S
= {1, 2, 3, 4, 5, 6}
You win
if the
event
A
= {1, 2, 3}
occurs.
Now suppose that the die has been tossed and while you
are not told what face shows, you are told the number is
odd; i.e., that the
event
B
= {1, 3, 5}
has occurred.
Would you still say
P
(win) = 0.5?
Equivalently, would both
players still be willing to bet even money?
=
2
1
6
3
=
=
S
A
)
(
win)
(
=
A
P
P
3
Recall that in the unconditional case (no additional knowledge):
Sample Space:
S
= {1, 2, 3, 4, 5, 6}
You win
if the
event
A
= {1, 2, 3}
occurs.
2
1
6
3
)
(
win)
(
=
=
=
=
S
A
A
P
P
Game:
Toss a die and observe the top face
Revised Sample Space:
B
= {1, 3, 5}
You win
if the
event
A
∩
B
= {1, 3}
occurs.
3
2
=
∩
B
B
A
)
(
odd)

win
(
=
=
B
A
P
P
4
The conditional probability formula
that we have derived works only for
finite sample spaces in which all
outcomes are equally likely (a
uniform discrete probability model
).
B
B
A
B
A
P
∩
=
)
(
Dividing both numerator and denominator of the fraction
by 
S
 does not change its value and gives us a formula
which we use for the definition of
conditional probability
:
)
(
)
(
)
(
B
P
B
A
P
S
B
S
B
A
B
B
A
B
A
P
∩
=
∩
=
∩
=
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5
)
(
)
(
)

(
B
P
B
A
P
B
A
P
∩
=
Definition: The conditional probability
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