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The Monty Hall Problem

The television game show Let's Make a Deal first aired in 1963. Host Monty Hall would give contestants a chance to win a modest prize, then a chance to trade their prize for a better prize. In one scenario Hall presented the contestand with three doors. Behind one of the doors was a big prize, and behind the other two were Zonks, items of little or no value.

In the late 1980s Marilyn vos Savant was the Guinness World Record holder for the highest IQ and the author of the "Ask Marilyn" column for the weekly Parade magazine, in which she provided a definitive answer for her audience asked.

One answer from vos Savant's column of September 9, 1990 caused a firestorm in which more than 10,000 readers (including many with PhD degrees) complained that she was wrong. Craig Whitaker of Columbia, Maryland asked a question that has become known as the Monty Hall Problem due to its similarity to the typical Let's Make a Deal scenario. Whitaker's question was:

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?

Vos Savant answered, yes, it is to your advantage to switch. Her reasoning was that there's a 1/3 chance that the car is behind the door you picked, and a 2/3 chance that it's behind another door. So by switching, you should win 2/3 of the time.

This answer is not intuitive, and vos Savant received so much criticism and hate mail that she spent three subsequent columns attempting to to explain and clarify her answer. To this day there are people with advanced math degrees who still think she gave the wrong answer, but a number of objective studies have proved she was right.

Perhaps it's easier to understand by looking at what's happening when the host opens a door. If the car is not behind the door you've picked, the host is actually revealing the location of the car by opening the door with a goat. That is, if you pick door 1, you'll know the car is behind door 2 if the host opens door 3, or it's behind door 3 if he opens door 2. In either case, you're guaranteed to win by switching. It is only if you've happened to pick the door with the car on the first try that opening another door does not reveal the car's location. And that's the one time you'll lose by switching. So, in the 1/3 of games that you unknowingly pick the car on your first guess, you'll lose by switching, and in the other 2/3 you'll will by switching.

Even for many people with advanced math degrees, it's still not intuitive. This may be due to the ambiguity of the scenario. The key is that the host always opens a door to reveal a goat. Although the contestant's original guess is random, the opening of the first door is not. It might become more clear if we added more doors.

Suppose you're on a game show, and you're given a choice of 100 doors. Behind one door is a car, and behind the other 99 are goats.  You pick a door, say number 1. The host, who knows what's behind each door, opens doors 2 through 62 and 64 through 100, revealing goats behind all of them. The host asks if you'd like to stick with door 1, or switch to door 63. In this scenario, most people's intuition leads them to the correct answer, that there's a 99% likelihood the car is behind door 63.

What may not be obvious is how the three-door game and the 100-door game are identical. In both cases, you're given the option of staying with your initial pick, or choosing any other door that might be hiding the car. We could rephrase the three-door scenario this way: After you pick a door, the host offers you a choice between keeping whatever is behind that door, or switching to whatever is behind the other two doors. But if you switch, you'll have to give the host a goat. The host then opens a door to reveal the goat you'll be returning to him if you switch. If switching gives you two doors instead of one, it should be more clear that switching gives you a better chance of winning.

But if you're still not convinced, you can try it yourself. I've created a JavaScript version of the game, where you can try to outwit the host and win the car. With a small number of trials anything can happen, but if you give it around 15-20 attempts with each strategy, the benefits of switching should become obvious.

 

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