A group of us are taking a class from a teacher who is trying to get us to understand the Monty Hall game. In the game, Monty shows us three doors. He promises that there is a good prize behind one, a pig behind one and nothing behind the third. Monty asks us to select a door. Then, he opens one of the other doors and he always selects an empty door. Then, he asks us to make a choice: either the initial door we selected or the remaining door that has not been opened.
Our teacher keeps explaining why we would win 2/3 of the time if we switch our choice from the initial door to the remaining door. He calls that "switching" and he calls staying with our original choice, "sticking".
This is a very famous problem and nearly everyone, professional probabilist or not, gets it wrong. Nearly everyone thinks that the chances of finding the prize at first or behind the door Monty doesn't open for us are equal. The writer Marilyn Vos Savant explained the solution in her column in Parade magazine and got all sorts of negative reactions from people who said she had made a mistake.
It is true that if the prize is put behind one of three doors at random, I have a 1/3 chance of picking the correct door. Put another way, there is a 2/3 chance that it is somewhere else and not behind the door I selected. The way the game is played, Monte shows me which of the other doors doesn't have the prize. Thus, if I switch to the door not yet chosen or opened, I have a 2/3 chance of getting the prize.
Putting "Monty Hall problem" or "Monty Hall simulation" into Google will take you to discussion, analysis and simple simulations of the game. You can play repeatedly and see what happens.
