What Is the Monty Hall Problem? The Puzzle That Broke Everyone's Intuition

The overwhelming instinct is that once one door is eliminated there are two doors left, so the odds must be even, and staying or switching cannot matter. The error is in treating the host's action as if it were random. It is not: the host knows where the car is and will never open the door hiding it, which means the door he opens carries information about where the car is not. Your original pick was made when you knew nothing, so it had a one in three chance of being right, and nothing that happens afterwards changes that, because the host will always be able to open a goat door whatever you picked. The probability that the car is behind one of the two doors you did not pick was two in three at the start, and since the host has now shown you which of those two it is not, all of that two in three has collected on the single remaining door.

The clearest demonstration is simply to list what happens in each case, assuming you always switch:

• •You picked the car, probability one in three. The host opens either goat door, you switch, and you get a goat. You lose
• •You picked goat one, probability one in three. The host must open the other goat door, you switch to the remaining door, and you get the car. You win
• •You picked goat two, probability one in three. The same applies, and you win
• •So switching wins in two of the three equally likely starting cases, and staying wins in one
• •An alternative that convinces almost everyone: imagine a hundred doors, you pick one, and the host opens ninety-eight goat doors. The remaining door suddenly looks very attractive, and the logic is identical

The result depends on assumptions that the puzzle usually states loosely, and changing them changes the answer, which is part of why the argument was so heated. The host must know where the car is; if he opens a door at random and it happens to show a goat, the odds really are even and switching gains nothing. He must always open a door and always offer the switch, rather than doing so only when you have picked the car, in which case switching would be a trap. And he must choose randomly between the two goat doors when you have picked the car, since a host with a predictable preference leaks additional information. On the actual American television show that gave the problem its name, the host did not follow these rules consistently, and Monty Hall himself pointed out in an interview that he was under no obligation to offer the switch at all, which makes the real game a psychological one rather than a probability exercise.

Marilyn vos Savant answered the question in her Parade magazine column and set off one of the largest public mathematical disputes on record. The volume of hostile mail was extraordinary, roughly a thousand letters came from people with doctorates, and several were published, including one from a mathematician telling her that she was the goat and another warning that the country already had enough innumeracy. She was right, the correspondents were wrong, and she went on to recommend that schools run the experiment physically, which thousands did, producing the two-thirds result and ending most of the local arguments. Paul Erdos, one of the most prolific mathematicians of the century, reportedly refused to accept the answer until he was shown a computer simulation. The episode is now a standard case study in the psychology of probabilistic reasoning, and the more interesting question is not why lay readers got it wrong but why so many trained mathematicians did.

Psychologists have studied the puzzle extensively because it isolates a specific failure. Part of it is the equiprobability bias, a strong default assumption that two remaining options must be equally likely, which people apply even when the setup clearly says otherwise. Part is a failure to condition on how information arrived, since the same observation carries different weight depending on the process that produced it, which is the error that Bayes's theorem is designed to prevent and which appears throughout medical testing and legal reasoning. And part is that people treat their own initial choice as if it were privileged, an endowment effect applied to a guess. Experiments show that most people continue to choose wrongly even after the explanation, that repeated play improves performance slowly, and that pigeons, tested with an equivalent apparatus, converge on the optimal strategy faster than humans do, apparently because they respond to outcome frequencies rather than to a theory of the situation.

Switching wins two times in three because the original choice was made with a one-in-three chance and nothing afterwards changes it, while the host's knowledge concentrates the remaining two-thirds onto the single unopened door. The result depends on the host knowing where the car is, always opening a goat door and always offering the switch. A magazine column giving the correct answer in 1990 drew around ten thousand objections, many from mathematicians, and the puzzle is now a standard case in the psychology of probability.