# The Monty Hall Problem

on December 1, 2012 with

Recently, FedSmith.com ran an article on three brainteasers for making meetings more interesting. Here is one that has been around, in one form or another, for many years. Be advised, though: post it at the beginning of the meeting, but don’t provide the answer until the meeting is over. Otherwise, there will be disruption, and the business of the meeting might be jeopardized!

There are three closed doors. Monty Hall, your host, tells you one door hides a wonderful prize and there is a goat behind each of the other two doors. Pick one.

You pick a door. Monty then opens one of the remaining doors and shows you a goat. He offers to let you trade the door you selected for his unopened door. Do you do it? Why?

Typically, players reason there is one prize behind one of the two doors that are left. The chances are 50:50 for either door. Makes no difference whether you change doors. Is this your choice? Are you sure?

The correct answer is yes, you trade your door for Monty’s. You do this because then you have a better chance of winning the prize.

## Explanation of the Answer to the Monty Hall Problem

Most people get this one wrong. Marilyn vos Savant, who featured the Monty Hall problem in her nationally syndicated “Parade” column, reported that 92% of her readers got it wrong. Even Paul Erdos, one of the greatest mathematicians of the last century, swore Marilyn’s answer could not be right. However, when shown a computer simulation, Erdos changed his mind.

After you select a door and before Monty has opened one of his, his chances of winning are twice as good as yours, because he has two doors, or chances, while you have just one. Yes?

There is just one prize, so at least one of Monty’s two doors must have a goat behind it. This is true, is it not?

If you agree that Monty must have a goat behind one of his doors, then you must also admit that by showing you an open door with a goat he has not proved anything – he has not changed the odds. Monty is still more likely than you to have the prize.

So, with Monty more likely to have the door with the prize, when he asks you if you want to trade doors, you say yes! Are you convinced? The Monty Hall problem appears to be a fine example of “thinking outside the box,” a skill we all need, especially Feds.