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Events and Probability Spaces
14.1
Let’s Make a Deal
In the September 9, 1990 issue of
Parade
magazine, columnist Marilyn vos Savant
responded to this letter:
Suppose you’re on a game show, and you’re given the choice of three
doors. Behind one door is a car, behind the others, goats. You pick a
door, say number 1, and the host, who knows what’s behind the doors,
opens another door, say number 3, which has a goat. He says to you,
”Do you want to pick door number 2?” Is it to your advantage to
switch your choice of doors?
Craig. F. Whitaker
Columbia, MD
The letter describes a situation like one faced by contestants in the 1970’s game
show
Let’s Make a Deal
, hosted by Monty Hall and Carol Merrill. Marilyn replied
that the contestant should indeed switch. She explained that if the car was behind
either of the two unpicked doors—which is twice as likely as the the car being
behind the picked door—the contestant wins by switching. But she soon received
a torrent of letters, many from mathematicians, telling her that she was wrong. The
problem became known as the
Monty Hall Problem
and it generated thousands of
hours of heated debate.
This incident highlights a fact about probability: the subject uncovers lots of
examples where ordinary intuition leads to completely wrong conclusions. So until
you’ve studied probabilities enough to have refined your intuition, a way to avoid
errors is to fall back on a rigorous, systematic approach such as the Four Step
Method that we will describe shortly. First, let’s make sure we really understand
the setup for this problem. This is always a good thing to do when you are dealing
with probability.
14.1.1
Clarifying the Problem
Craig’s original letter to Marilyn vos Savant is a bit vague, so we must make some
assumptions in order to have any hope of modeling the game formally. For exam
ple, we will assume that:
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Chapter 14
Events and Probability Spaces
1. The car is equally likely to be hidden behind each of the three doors.
2. The player is equally likely to pick each of the three doors, regardless of the
car’s location.
3. After the player picks a door, the host
must
open a different door with a goat
behind it and offer the player the choice of staying with the original door or
switching.
4. If the host has a choice of which door to open, then he is equally likely to
select each of them.
In making these assumptions, we’re reading a lot into Craig Whitaker’s letter. Other
interpretations are at least as defensible, and some actually lead to different an
swers. But let’s accept these assumptions for now and address the question, “What
is the probability that a player who switches wins the car?”
14.2
The Four Step Method
Every probability problem involves some sort of randomized experiment, process,
or game. And each such problem involves two distinct challenges:
1. How do we model the situation mathematically?
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Probability, Probability theory, Probability space, Monty Hall, monty hall problem

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