I was so taken with Leonard Mlodinow‘s new book The Upright Thinkers, that I went on Amazon and ordered some of his previous books, including The Drunkard’s Walk: How Randomness Rules Our Lives. In an early chapter discussing the mathematical development of probability, Mlodinow discusses the Monty Hall problem, made famous in 1990 by Marilyn vos Savant in Parade magazine. In her Ask Marilyn column, reproduced (with slight modifications) by Mlodinow, Savant was asked:
Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, “Do you want to switch to the other unopened door?” Is it to the contestant’s advantage to make the switch?
Marilyn answered “yes,” but thousands of her readers, including some with advanced degrees, wrote her to tell her she made a mistake. Marilyn solved the problem using elementary probability. If you are given a choice of three doors, you will pick the correct one just 1/3 of the time. For the 2/3 of the time that you initially chose incorrectly, you could win by switching your choice to the door that remains after the host reveals the first goat. This seems a bit counterintuitive, which is why so many of her readers were upset, because you are still left choosing between two doors, not knowing what is behind either one. It seems like a 50-50 choice.
Many people, including Mlodinow, use this example to show how our intuition can be faulty and not correct according to the laws of probability. Daniel Kahneman won a Nobel Prize in Economics for this type of work. But the problem with this example (and even some of Kahneman’s that I have read) is that they don’t really prove that our intuition is wrong.
Take the Monty Hall problem. Look at it more closely and consider this: how does the contestant know what game he is playing? All he knows is that he was asked to choose one of three doors, then one of the unchosen doors was exposed and he was asked whether he wanted to switch. That’s it. The contestant has no way of knowing what game the host is actually playing. And that makes all the difference for what our intuition tells us to do.
It isn’t hard to imagine that this is not the game that Marilyn thought it was. What if the game was this: if you choose incorrectly, the host opens the door to reveal your goat; but if you choose correctly, the host tries to induce you to change until you pick a door that has a goat or accept a smaller gift. Sometimes he opens one of the unopened doors to reveal a goat and asks if you want to switch to the other unopened door (as in Marilyn’s example). Sometimes he asks you if you want cash instead of what is behind the door you chose. Etc. Etc. You never know if or how the game will end. Now this game looks to you the contestant exactly like the game that Marilyn described. If the host induces you to switch your choice, it means you’ve lost. Do you want to follow Marilyn’s advice? You’d lose every single time.
So, back to our supposedly faulty intuition. Is it really so bad? What makes more sense: that the host would give us an easy chance to double our odds of winning, or that he would try to trick us into giving up our correct choice? Our intuition tells us to be skeptical. It tells us that we don’t have enough information to know what game we are playing, and so the choice is really 50-50. We just don’t know if the information we received when the host revealed the goat really means anything. So we are left again with choosing randomly between two doors.
While it is certainly true that our intuition often is at odds with the laws of probability, (and indeed that is why many people are initially confused by this problem), it turns out that the confusion is sometimes more warranted than strict probabilists are willing to admit. There are plenty of real-world examples that show how bad people are at probability. Take, for example, the fact that so many people buy lottery tickets, which are a terrible bet that no one in his right mind would take. (In other words, the expected payoff is far less than the cost of a ticket.) It makes you wonder though, why so many smart people like to use completely unreal problems, like the Trolley problem, or flawed ones like the Monty Hall problem, to make the same point. You really cannot test the correctness of our real-world intuition with unreal make-believe problems where people can’t even know what game they are playing.
[For a more thorough enumeration of how Marilyn might have been wrong when answering this problem, see Herb Weiner’s Marilyn is tricked by a game show host.]