The Monty Hall Problem

In 1990 Craig Whitaker of Columbia Maryland asked Marilyn Vos Savant the following:

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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

From Leonard Mlodinow in The Drunkard's Walk:

It appears to be a pretty silly question. Two doors are available — open one and you win; open the other and you lose — so it seems self-evident that whether you change your choice or not, your chances of winning are 50/50. What could be simpler? The thing is, Marilyn said in her column that it is better to switch.

Despite the public’s much-heralded lethargy when it comes to mathematical issues, Marilyn’s readers reacted as if she’d advocated ceding California back to Mexico. Her denial of the obvious brought her an avalanche of mail, 10,000 letters by her estimate. If you ask the American people whether they agree that plants create the oxygen in the air, light travels faster than sound, or you cannot make radioactive milk by boiling it, you will get double-digit disagreement in each case (13 percent, 24 percent, and 35 percent, respectively). But on this issue, Americans were united: Ninety-two percent agreed Marilyn was wrong.

Martin Gardner's Three Prisoners Problem:

Three prisoners, A, B and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the others who is going to be executed. "If B is to be pardoned, give me C's name. If C is to be pardoned, give me B's name. And if I'm to be pardoned, flip a coin to decide whether to name B or C."

The warden tells A that B is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and C. Prisoner A secretly tells C the news, who is also pleased, because he reasons that A still has a chance of 1/3 to be the pardoned one, but his chance has gone up to 2/3. What is the correct answer?

A problem with money:

You know that your pocket contains either a $1 bill or a $20 bill (each is equally likely) and no other bills. A friend then gives you a $1 bill which you put in your pocket. Later you take a random bill from your pocket and discover that it is a $1. What is the probability that the other bill in your pocket is a $20?