The Monty Hall Problem

Monty Hall hosted a game show around 25 years ago in which he fast-talked people into making deals for cash and prizes. Audience members wore ridiculous costumes, and at the end of the show, Monty would wander through the melange of clowns, chickens, and chefs, offering quick cash if selected people could produce a desired item.

"I'll give you $40 if you have hairspray in your purse," Monty would ask some woman in a green bodysuit with foam flower petals surrounding her head. It was a laugh riot.

The centerpiece of the show, however, was "The Deal". Monty would pick someone from the audience, give them cash, the tell them that behind one of three doors was a lovely gift. They could keep the cash, or pick one of the doors.

After picking, say, door #1, the real fun began. Without opening door #1, Monty would reveal the contents of door two, usually a lovely Amana kitchen appliance. Then he would ask if they wanted to switch from door #1, contents unknown, to the remaining door, #3.

This has come to be known as the Monty Hall Problem. It is very well known in mathematical and statistical circles because the correct answer to the problem is counterintuitive. You should always switch.

It seems like all doors have an equal chance of holding the prize, 1/3, hence switching gains nothing. Not true. Switching increases the chance to 2/3.

There are many web sites dedicated to demonstrating this. I also found a java applet that illustrates the problem.

The discussions you find above will show outcome charts, use probability buzzwords, and often overanalyze the problem. The best explanation I know, however, comes from a friend of mine, Jake Schmidt. It is simple, easy to grasp, and obtains the correct numerical answer.

Ignore for a moment that Monty opens up one of the remaining doors after you make the initial selection. Realize that what he is doing, in reality, is offering to trade you both of the remaining doors for the one you first chose. This exchange increases your chances of winning from 1/3 to 2/3.

By opening one of the doors, Monty just confuses you into believing this is a one-for-one trade. There is no doubt when the problem is put in these terms. The problem is only a problem because of Monty's distraction.

Lovely.

Incidentally, people also seem to be in love with gameshows. Like I said, Americans can be crazy.

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Last Updated: 9 July 1997
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