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# sci.math FAQ: Monty Hall problem

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```Archive-Name: sci-math-faq/montyhall
Last-modified: December 8, 1994
Version: 6.2

The Monty Hall problem

This problem has rapidly become part of the mathematical folklore.

The American Mathematical Monthly, in its issue of January 1992,
explains this problem carefully. The following are excerpted from that
article.

Problem:

A TV host shows you three numbered doors (all three equally likely),
one hiding a car and the other two hiding goats. You get to pick a
door, winning whatever is behind it. Regardless of the door you
choose, the host, who knows where the car is, then opens one of the
other two doors to reveal a goat, and invites you to switch your
choice if you so wish. Does switching increases your chances of
winning the car?

If the host always opens one of the two other doors, you should
switch. Notice that 1/3 of the time you choose the right door (i.e.
the one with the car) and switching is wrong, while 2/3 of the time
you choose the wrong door and switching gets you the car.

Thus the expected return of switching is 2/3 which improves over your
original expected gain of 1/3 .

Even if the hosts switches only part of the time, it pays to switch.
Only in the case where we assume a malicious host (i.e. a host who
entices you to switch based in the knowledge that you have the right
door) would it pay not to switch.

References

L. Gillman The Car and the Goats American Mathematical Monthly,
January 1992, pp. 3-7.

_________________________________________________________________

alopez-o@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995

```