Page 193 - Probability Demystified

182 CHAPTER 10 Simulation

The theoretical average or expected value can be found by using the
formula shown in Chapter 5.
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EðXÞ¼ ð$1Þþ ð$5Þþ ð$10Þ¼ $3:83. Actually, I did somewhat better
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than average.
The Monty Hall problem is a probability problem based on a game played
on the television show ‘‘Let’s Make A Deal,’’ hosted by Monty Hall.
Here’s how it works. You are a contestant on a game show, and you are to
select one of three doors. A valuable prize is behind one door, and no prizes
are behind the other two doors. After you choose a door, the game show host
opens one of the two doors that you did not select. The game show host
knows which door contains the prize and always opens a door with no prize
behind it. Then the host asks you if you would like to keep the door you
originally selected or switch to the other unopened door. The question is ‘‘Do
you have a better chance of winning the valuable prize if you switch or does it
make no difference?’’
At first glance, it looks as if it does not matter whether or not you
switch since there are two doors and only one has the prize behind it. So
the probability of winning is 1 whether or not you switch. This type
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of reasoning is incorrect since it is actually better if you switch doors!
Here’s why.
Assume you select door A. If the prize is behind door C, the host opens
door B, so if you switch, you win. If the prize is behind door B, the host
opens door C, so if you switch, you win. If the prize is behind door A and no
matter what door the host opens, if you switch, you lose. So by always
switching, you have a 2 chance of winning and a 1 chance of losing. If you
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don’t switch, you will have only a chance of winning no matter what. You
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can apply the same reasoning if you select door B or door C. If you always
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switch, the probability of winning is . If you don’t switch, the probability of
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winning is .
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You can simulate the game by using three cards, say an ace and two kings.
Consider the ace the prize. Turn your back and have a friend arrange the
cards face down on a table. Then select a card. Have your friend turn over
one of the other cards, not the ace, of course. Then switch cards and see
whether or not you win. Keep track of the results for 10, 20, or 30 plays.
Repeat the game but this time, don’t switch, and keep track of how many
times you win. Compare the results!
You can also play the game by visiting this website:
http://www.stat.sc.edu/ west/javahtm/LetsMakeaDeal.html