16.1: Let's Make a Deal

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Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer
Google and Massachusetts Institute of Technology via MIT OpenCourseWare

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In the September 9, 1990 issue of Parade magazine, columnist Marilyn vos Savant responded to this letter:

\(\nonumber \textit{Suppose you’re on a game show, and you’re given the choice of three} \\ \textit{doors. Behind one door is a car, behind the others, goats. You pick a} \\ \textit{door, say number 1, and the host, who knows what’s behind the doors,} \\ \textit{opens another door, say number 3, which has a goat. He says to you,} \\ \textit{“Do you want to pick door number 2?” Is it to your advantage to} \\ \textit{switch your choice of doors?}\)

\(\nonumber \text{Craig. F. Whitaker} \\ \text{Columbia, MD}\)

The letter describes a situation like one faced by contestants in the 1970’s game show Let’s Make a Deal, hosted by Monty Hall and Carol Merrill. Marilyn replied that the contestant should indeed switch. She explained that if the car was behind either of the two unpicked doors—which is twice as likely as the the car being behind the picked door—the contestant wins by switching. But she soon received a torrent of letters, many from mathematicians, telling her that she was wrong. The problem became known as the Monty Hall Problem and it generated thousands of hours of heated debate.

This incident highlights a fact about probability: the subject uncovers lots of examples where ordinary intuition leads to completely wrong conclusions. So until you’ve studied probabilities enough to have refined your intuition, a way to avoid errors is to fall back on a rigorous, systematic approach such as the Four Step Method that we will describe shortly. First, let’s make sure we really understand the setup for this problem. This is always a good thing to do when you are dealing with probability.

Clarifying the Problem

Craig’s original letter to Marilyn vos Savant is a bit vague, so we must make some assumptions in order to have any hope of modeling the game formally. For example, we will assume that:

The car is equally likely to be hidden behind each of the three doors.
The player is equally likely to pick each of the three doors, regardless of the car’s location.
After the player picks a door, the host must open a different door with a goat behind it and offer the player the choice of staying with the original door or switching.
If the host has a choice of which door to open, then he is equally likely to select each of them.

In making these assumptions, we’re reading a lot into Craig Whitaker’s letter. There are other plausible interpretations that lead to different answers. But let’s accept these assumptions for now and address the question, “What is the probability that a player who switches wins the car?”