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Unformatted text preview: 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(Ac ∩ B ) = P(B  A)P(A) + P(B  Ac )P(Ac ).
5
7
The probability of A is 12 and the probability for Ac is 12 . Given that the
4
ﬁrst ball is red, there are now 4 red balls and 7 green, so P(B  A) = 11 .
5
Similarly, P(B  Ac ) = 11 . Therefore P(B ) = 57
5
45
·
+
·
=,
11 12 11 12
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
switching to the remaining door. Should you do it?
Answer. Let’s suppose you choose door 1, since the same analysis applies
whichever door you chose. Strategy one is to stick with door 1. With probability 1/3 you chose the car. Monty Hall shows you one of the other doors.
Since you know he won’t show you the door with the car, your probability is
still 1/3.
Strategy 2 is to change. With probability 1/3 you chose the car. He shows
you another door, say, door 2, with straw, and then you choose door 3 and
lose. With probability 2/3 the car is behind door 2 or 3. He shows you the
door the car is not behind, say, door 2, and you change to door 3, and win.
So switching increases your probability of winning to 2/3.
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan
 Probability

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