This is a problem which has cropped up more than once on The Problem Site and its sister sites. The question is this: 

Suppose you are on a game show, and Monty Hall (the game show host) shows you three doors, and tells you that there is a prize behind only one of them. He asks you to choose one door.

Once you've chosen a door, Monty (who knows where the prize is) opens one of the other doors to show you that it doesn't contain a prize. He then asks you if you want to change your guess (to the other door which remains closed) or keep your guess.

Which is the wisest course of action? 
 

Solution

Believe it or not, you are better off changing your guess. Your probability of winning by changing your guess is 2/3, as opposed to 1/3 probability of winning by keeping your guess the same.

This is very counterintuitive, and many people don't believe it - even after seeing a rigorous mathematical proof. There are sites out there that explain in great detail why you're better off changing your guess, and still people don't believe it. At the bottom of this page you'll find a link to a simulation which you can try it out for yourself.

But first, here is an explanation that may help you believe. It is not a rigorous mathematical proof, but an extreme example that'll help you believe. This explanation serves as a good reminder that sometimes generalizing or creating extreme examples can help us revisualize a problem.

Suppose Monty had shown you a million doors, and asked you to pick the correct one. When you make your guess, you know your chance of winning is (quite literally) one in a million! You're not at all optimistic about winning. In fact, you are 99.999999% sure that you're wrong!

Now Monty (who knows where the prize is) goes down the line and opens every door you didn't select except one. What does that tell you? It tells you that if you guessed wrong, then the one remaining door that's closed must be the one with the prize. The odds? 999,999/1,000,000! (That's one minus the probability that you guessed correctly in the first place!)

You'd have to be crazy not to change your guess under those circumstances! Now apply that same reasoning to the three door problem, and you'll see that you're better off changing your guess.

Isn't that slick?

Try the Monty Hall Simulation!

1.

If you play the game 3000 times, and you change your guess each time, about how many times would you expect to win?

2.

If you play the game 1000 times, and never change your guess, about how many times would you expect to win?

3.

Suppose you played 3 times, and only won once by changing your guess each time. Does that prove that the probability of winning is 1/3?