### Monty Hall Problem

The Monty Hall problem is a counter-intuitive situation made popular by the T.V. game show *Let's Make A Deal*. The siuation was described in *Parade* magazine:

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? (Whitaker, Craig F. (1990). [Letter]. "Ask Marilyn" column,Parade Magazinep. 16 (9 September 1990).)

A complete and overly pedantic discussion is given in Wikipedia: Monty Hall Problem, but a simpler analysis is possible by calculating the expectation. Everyone should agree that the probability of initially choosing the winning door is 1/3. Let us give the winning door a value of 1 and the other two doors a value of 0. Then the expectation of the initial choice is:

E_{initial} = 1·1/3 + 0·2/3 = 1/3

If we decide not to switch doors after being shown an empty door, then this remains the expectation, since it is **always** possible for the host to expose an empty door regardless of whether our initial choice was correct or incorrect -- i.e., exposing the other door provides no new information:

E_{stay} = 1·1/3 + 0·2/3 = 1/3

(Some think that because there are only two choices left, the probability has increased to 1/2, but this is an error because the situation has not changed, so the probability cannot have changed.)

Now consider the expectation if we switch. We will have chosen the correct door 1/3 of the time, so in switching we will switch to a worthless door, but we will have chosen a worthless door 2/3's of the time, so in switching we switch to the correct door (since the other worthless door has been exposed), and the expectation for a switching strategy is:

E_{switch} = 0·1/3 + 1·2/3 = 2/3

Switching has doubled the probability of choosing the correct door -- this is a bet I would take anytime.

The reaction to this correct answer being published in *Parade* was surprising:

When the problem and the solution appeared inParade, approximately 10,000 readers, including nearly 1,000 with Ph.D.s, wrote to the magazine claiming the published solution was wrong.

…

When first presented with the Monty Hall problem an overwhelming majority of people assume that each door has an equal probability and conclude that switching does not matter (Mueser and Granberg, 1999). Out of 228 subjects in one study, only 13% chose to switch (Granberg and Brown, 1995:713). In her bookThe Power of Logical Thinking, vos Savant (1996:15) quotes cognitive psychologist Massimo Piattelli-Palmarini as saying "... no other statistical puzzle comes so close to fooling all the people all the time" and "that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer."

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