Marilyn vos Savant has the highest IQ ever measured. Savant was asked the following question in her September 9, 1990 column. Her answer is wrong.

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?

— Craig F. Whitaker Columbia, Maryland, [24]

This question is called the Monty Hall problem due to its resembling scenarios on the game show Let’s Make a Deal; its answer existed before it was used in “Ask Marilyn”. She said the selection should be switched to door #2 because it has a 2/3 chance of success, while door #1 has just 1/3. To summarize, 2/3 of the time the opened door #3 will indicate the location of the door with the car (the door you had not picked and the one not opened by the host). Only 1/3 of the time will the opened door #3 mislead you into changing from the winning door to a losing door. These probabilities assume you change your choice each time door #3 is opened, and that the host always opens a door with a goat. This response provoked letters from thousands of readers, nearly all arguing doors #1 and #2 each have an equal chance of success. A follow-up column reaffirming her position served only to intensify the debate and soon became a feature article on the front page of The New York Times. Parade received around 10,000 letters from readers who thought her wrong.

Under the “standard” version of the problem, the host always opens a losing door and offers a switch. In the standard version, Savant’s answer is correct. However, the statement of the problem as posed in her column is ambiguous. The answer depends on what strategy the host is following. If the host operates under a strategy of only offering a switch if the initial guess is correct, it would clearly be disadvantageous to accept the offer. If the host merely selects a door at random, the question is likewise very different from the standard version. Savant addressed these issues by writing the following in Parade Magazine, “the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. Anything else is a different question.”

She expounded on her reasoning in a second follow-up and called on school teachers to show the problem to classes. In her final column on the problem, she gave the results of more than 1,000 school experiments. Nearly 100% found it pays to switch. Of the readers who wrote computer simulations of the problem, 97% reached the same conclusion. Most respondents now agree with her original solution, with half of the published letters declaring their authors had changed their minds.

Here’s why her idea is wrong. Regardless of the first choice, the initial choice has a 1 in 3 chance of being correct regardless of the door that’s chosen. Doors 1,2 and 3 each have one chance in three of being correct. The sum of the choices is 3 times one out of three, a total of three out of three chances that one choice is correct. When one door is opened the original problem is over. The chance of winning the second round becomes one out of two but either choice has the identical chance of being correct. There is no advantage to changing the original choice. There is no disadvantage to changing either.

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