## Monday, February 23, 2015

The famous Monty Hall problem has been analyzed in great detail, but those new to it still get the answer wrong, even some professional mathematicians. I’ve had several arguments about it, but I’ve found only one way to convince people of the correct answer: offer to bet.

Here’s a version of the Monty Hall problem:
You’re on a game show. There are three doors. You’re told there is a car behind one door and a goat behind each of the other two doors. You’re asked to pick the door you think has the car. After you choose, the game show host, Monty Hall, opens one of the other doors to reveal a goat. Monty then asks you if you want to stick with your door or whether you want to switch to the other unopened door. Should you switch?
To make things more precise, here’s some additional information: before the game began, Monty intended to open a door you didn’t pick to show a goat (no matter what door you chose), and he intended to offer you a chance to switch.

It’s well known that the probability of winning the car is one-third if you don’t switch and two-thirds if you do switch. However, a great many people feel certain that the odds are 50-50 either way. The challenge is to convince them they’re wrong. The only way I’ve ever had success, particularly when the argument has an audience, is to offer to bet.

Suppose that Frank believes the odds are 50-50 and wants to bet with me. To simulate the game, we appoint a neutral party to play the role of Monty. Monty rolls a fair die where he can see it but nobody else can see it. The outcomes 1 and 2 mean the car is behind the first door, 3 and 4 the second door, and 5 and 6 the third. I let Frank pick a door. Then Monty tosses a fair coin (whose outcome only he can see) to choose which other door to open. The coin toss is only needed when Frank chose the correct door, but Monty tosses the coin every time to avoid revealing information. Then Monty reveals the die to show us whether Frank picked the right door.

If Frank is right about the odds, he will have the correct door half the time. If I’m right, Frank will have the correct door one-third of the time. Based on Frank’s 50-50 odds, it would be fair to bet \$2 on each play. Based on one-third and two-thirds, it would be fair for Frank to pay me \$2 when he has the wrong door and for me to pay Frank \$4 when he has the right door. To split the difference, we make it so that I win \$2 when Frank has the wrong door, and I lose \$3 when Frank has the right door. This way, both Frank and I believe we’ll make money over time.

The goal with all this talk of betting isn’t to make any money. I’ve found that people like Frank start to have doubts when real money gets involved and they begin to think. They slowly realize that they only win if their original door choice matches the roll of the die, which will happen one-third of the time.

The first time I had a serious argument about Monty Hall, I was a summer student who was too foolish to realize it’s a bad idea to make a senior engineer look foolish over trivial matters. The talk of betting ended the public argument with “Frank,” and I got a private admission a few hours later that he was wrong. I wasn’t very popular with Frank’s crowd the rest of that summer.

So, it’s probably a social mistake to argue about Monty Hall in the first place, but if you actually want to change someone’s mind, my experience has been that offering to bet on Monty Hall simulations is very effective.

1. I'll take door #1, no #2, no #3... no I'll take door #4! (Ode to George Carlin)

Interesting stuff,

2. That's a tricky one to reason about. I convince myself by breaking it down into two cases. 2/3 of the time I initially pick a goat, and switching gets me a car. 1/3 of the time I initially pick the car and switching gets me a goat. Have you ever tried that reasoning with skeptics?

1. @Greg: Skeptics have their faulty reasoning for the 50-50 answer and then they don't listen because they know they're obviously right. It takes talk of betting and the possibility of losing money to cause them to re-evaluate. Your reasoning should be quite sensible to anyone who will actually listen.

3. I already have a car. Come on goat! No whammies!

I think in the game, not so much in the analysis, there is also a commitment fallacy, and a loss-aversion fallacy. People probably tend to want to stick with their first guess, since they're lucky/on a roll. Also, maybe subconsciously we fear how we'll feel if we switch from the car to a goat.

I think the main thing is what you've detailed here though, that we don't think there's any point in switching, that the odds have remained the same. I remember reading the scenario the first time in a book and I was surprised by the result.

I'm writing this on Friday, so have a good weekend, Michael.

1. @Gene: The factors you mention could come into play if this game show were actually played. As I understand the history of Monty Hall, it was never actually played on a real game show. It's really just a probability question where a surprising number of mathematicians have a difficult time getting the right answer (or even believing it after they see it).