or
P
(
F
∩
E
) =
P
(
E
)
P
(
F
)
,
which agrees with the definition of independence we gave before.
Let us give two more examples.
An example: Suppose an urn holds 5 red balls and 7 green balls.
You
draw two balls without replacement. What is the probability the second ball
is red?
Answer. Let
A
be the event that the first ball is red,
B
that the second
ball is, and we want
P
(
B
). Then
P
(
B
) =
P
(
A
∩
B
) +
P
(
A
c
∩
B
) =
P
(
B

A
)
P
(
A
) +
P
(
B

A
c
)
P
(
A
c
)
.
The probability of
A
is
5
12
and the probability for
A
c
is
7
12
. Given that the
first ball is red, there are now 4 red balls and 7 green, so
P
(
B

A
) =
4
11
.
Similarly,
P
(
B

A
c
) =
5
11
. Therefore
P
(
B
) =
4
11
·
5
12
+
5
11
·
7
12
=
5
12
,
which is what one would expect.
An example. This is known as the Monty Hall problem after the host of
the TV show of the 60’s called
Let’s Make a Deal
.
There are three doors, behind one a nice car, behind each of the other
two a bale of straw. You choose a door. Then Monty Hall opens one of the
other doors, which shows a bale of straw. He gives you the opportunity of
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 Spring '10
 ansan
 Conditional Probability, Probability, Englishlanguage films, Duodecimal, Monty Hall, monty hall problem, Let's Make a Deal

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