# Marilyn is tricked by a game show host

Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 750 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."

The material from Marilyn's column quoted in this review originally appeared in Parade Magazine February 17, 1991, July 7, 1991, and at least one other issue. (Herb would appreciate copies of any earlier columns discussing this question.)

## Question

Suppose you're on a game show, and you're given a 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?

-- Craig F. Whitaker, Columbia MD.

Yes, you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here's a good way to visualize what happened. Suppose there are a million doors, and you pick door No. 1. Then the host, who knows what's behind the doors and will always avoid the one with the prize, opens them all except door No. 777,777. You'd switch to that door pretty fast, wouldn't you?

I'm receiving thousands of letters, nearly all insisting that I'm wrong. Of the letters from the general public, 92% are against my answer; and of the letters from universities, 65% are against my answer. Overall, nine out of 10 readers completely disagree with my reply.

## Sorry, Marilyn

Marilyn, there's nothing wrong with your math. As you noted, math answers aren't determined by votes. But TV ratings are! What could possibly have justified your assumption that the game show host offers every contestant the same choice? The initial question described only a single incident.

If I were the game show host, and you were the contestant, I'd offer you the option to switch only if you initially chose the correct door. In this case, the first door has a 100% chance of winning, the second door has a 0% chance, and switching would be a sure loser.

Unless you understand the motives and behavior of the game show host, all the mathematics in the world won't help you answer this question.

I assume that my letter to you explaining your incorrect assumption was lost in the noise of the thousands of letters you received.

## Clarification

Marilyn assumes that the gameshow host offers every contestant the opportunity to switch. In fact, there may be some game shows where this is the case. However, I believe that Marilyn was wrong to make such an assumption without stating it explicitly. If Marilyn had simply stated her answer was based upon the assumption that the game show host always offered the contestant the opportunity to switch, I would have been satisfied with her answer.

Assuming that the game show host does not offer this opportunity to every contestant, there are several possibilities:

• The host makes this offer only if the contestant is initially correct. In this case, switching would be a sure loser. Contestants would catch on pretty quickly to this, and nobody would ever switch.
• The host makes this offer only if the contestant is initially wrong. In this case, switching would be a sure winner. Since the contestant would always win, this would not make for a very exciting game show.
• The host makes this offer to selected contestants, for example, contestants that have not yet won any prizes. This would keep the show interesting, and would favor the underdog.
• etc.
For the sake of simplicity, I chose an extreme example to illustrate the error in Marilyn's analysis. I did not intend to imply that I would treat every contestant this way.

## An interesting alternative

Patrick J. LoPresti <patl@catfish.lcs.mit.edu> wrote to point out an interesting alternative. Although the original question states that the host knows what's behind the doors, he nevertheless has the option of randomly choosing which door to open second. This means that sometimes (but not in the example cited by the original question), the host will open the door containing the prize. Regardless of which door the contestant chooses, there are six equally likely possibilities. However, since the question states that the door contains a goat, two of these possibilities are ruled out by this additional information. Assuming that you pick door number 1, the six possibilities are:

• The prize is behind door 1, the host opens door 2, and you see a goat.
• The prize is behind door 1, the host opens door 3, and you see a goat.
• The prize is behind door 2, the host opens door 2, and you see a prize.
• The prize is behind door 2, the host opens door 3, and you see a goat.
• The prize is behind door 3, the host opens door 2, and you see a goat.
• The prize is behind door 3, the host opens door 3, and you see a prize.

Since the contestant saw a goat, the third and sixth possibilities are eliminated. The remaining for possibilities are equally likely. In two of these remaining four possibilities, you have already chosen the correct door, and switching would be a mistake. In the other two possibilities, switching would be a winner. Thus, switching would be a winner in two of the four possibilities, or 50%.

This confirms my original claim that unless you understand the motives and behavior of the game show host, there is no correct answer to this question.

It is interesting to analyze why the results suggested by Pat are different from Marilyn's results. Imagine that whenever the host randomly chooses the door containing the prize, before opening the door, he immediately chooses a different door. Now, the six equally likely possibilities are:

• The prize is behind door 1, the host chooses and opens door 2.
• The prize is behind door 1, the host chooses and opens door 3.
• The prize is behind door 2, the host chooses door 2, but opens door 3.
• The prize is behind door 2, the host chooses and opens door 3.
• The prize is behind door 3, the host chooses and opens door 2.
• The prize is behind door 3, the host chooses door 3, but opens door 2.

This makes switching a winner in four out of six possibilities, or 2 out of 3. This is the result reported by Marilyn. Although it appears as if there are only four possibilities (if one considers the third and fourth equivalent and the fifth and sixth equivalent), the four possibilities are not equally likely.

## Marilyn's Math is Corrrect!

I'm periodically receive email telling me about computer simulations proving that Marilyn is correct. (I never claimed that her math was wrong. I only challenge her assumptions, not her math.) I also get requests to explain why Marilyn's math is correct. So, assuming that the Host always offers the contestant a chance to switch, this is why Marilyn is correct:

After you pick but before you open any doors, there's a 1/3 chance that you've picked correctly, and a 2/3 chance that you've picked wrong. Assuming that the host can open doors, but can not move prizes, nothing that the host does will change the probabilities described above.

Now the host opens one of the doors, and there's nothing behind it. There's still a 1/3 chance that you've picked correctly, and a 2/3 chance that you've picked wrong. This means that the remaining door has a 2/3 chance of being correct.

I hope I've done a better job of explaining this than Marilyn.

## Monty Hall's Reaction

Thank you to Rick Merrill <rick@gcctech.com>, who found Behind Monty Hall's Doors: Puzzle, Debate and Answer? Here are some quotes from Behind Monty Hall's Doors:
• "So her answer's right: you should switch," Mr. Hall said.

• On the first, the contestant picked Door 1. "That's too bad," Mr. Hall said, opening Door 1. "You've won a goat." "But you didn't open another door yet or give me a chance to switch." "Where does it say I have to let you switch every time? I'm the master of the show. Here, try it again."

• You may think you have probability going for you when you follow the answer in her column, but there's the psychological factor to consider."

• Ms. vos Savant acknowledged that the ambiguity did exist in her original statement. She said it was a minor assumption that should have been made obvious by her subsequent analyses, and that did not excuse her professorial critics. "I wouldn't have minded if they had raised that objection," she said Friday, "because it would mean they really understood the problem. But they never got beyond their first mistaken impression. That's what dismayed me."

• Still, because of the ambiguity in the wording, it is impossible to solve the problem as stated through mathematical reasoning. "The strict argument, "Dr. Diaconis said, "would be that the question cannot be answered without knowing the motivation of the host."

• Which means, of course, that the only person who can answer this version of the Monty Hall Problem is Monty Hall himself. Here is what should be the last word on the subject:

"If the host is required to open a door all the time and offer you a switch, then you should take the switch," he said. "But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.

http://www.wiskit.com/marilyn.gameshow.html last updated January 1, 1997 by herbw@wiskit.com