The Monty Hall Paradox

For some reason, I’ve suddenly become crazy over randomness and probability. I’m only halfway through The Drunkard’s Walk but this will definitely be one of my favorite books.

Here’s the Monty Hall problem:

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 No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Marilyn vos Savant, famous for being listed for years in the Guinness World Records Hall of Fame as the person with the world’s highest recorded IQ (228), published this question in her column in Parade Magazine in September 1990. The question does appear quite silly. After all, when one door is shown to be a loser (assuming that we prefer a car to a goat), the probability of either remaining choice – neither which is more likely than the other – becomes 1/2. So why would switching make any difference? But Marilyn said in her column that it is better to switch.

This resulted in plenty of controversy, attracting some 10,000 mails including almost a thousand from PhDs. Many readers seemed to feel let down. How could a person they trusted so much on such a broad range of issues be confused by such a simple question?

When I was first asked the question, I too thought there would be no way that switching could make a difference. But the truth is, switching does increase your chances of winning. Here’s the solution from Wikipedia:

Singapore’s casino has officially opened – and never has the study of randomness been more timely.

Randomly yours,

Tim

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2 thoughts on “The Monty Hall Paradox

  1. using mathematical terms, can i say when i first choose a door, the initial probability of having a car behind was 1/3, but after another one has been revealed to be otherwise, the probability of having a car behind it has increased to 1/2. hence since the first door was chosen based on a 1/3 probabilty, it would be better to switch since the second door would have 1/2 chance of having a car behind it?

  2. If the probabilities do indeed change to 1/2, then it would make no difference whether or not you switch! But that’s incorrect.

    It’s easier if you think of it this way: say you choose Door A, your initial probability of getting it right is 1/3. Hence your probability of getting it wrong is 2/3, that is to say B or C has a 2/3 probability of being correct. By deliberately revealing a goat from Door B or C, the host intervenes in what has been a random process. Now the probability that the car is behind the remaining door increases to 2/3. Switching increases your probability of winning from 1/3 to 2/3.

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