The Birth of the Monty Hall Problem

The problem of doors, cars, and goats became famous through a newspaper column called Ask Marilyn. Marilyn is Marilyn vos Savant, known for being the smartest person in the world. In her column in September 1990, she posed a question something like this:
A contestant on a game show is given the choice of three doors: Behind one door is a car; behind the other two doors are goats. After the contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, and reveals a goat. He then says to the contestant, “Do you want to switch to the other unopened door?” Is it to the contestant’s advantage to switch doors?

The idea for the question stemmed from the TV game show Let's Make a Deal (which ran from 1963 to 1976) with host Monty Hall. Given its inspiration, this problem is often referred to as the Monty Hall Problem.

The Monty Hall Problem Evades Even the PhDs!

Upon seeing this question, Marilyn’s readers were in agreement—obviously if there were two doors left and one had a car behind it, there was a 50/50 chance of the car being behind each door. But when Marilyn posted that it was better to switch doors, chaos ensued. She estimated that she received 10,000 letters from her readers, including many PhDs, telling her that she was mistaken. But she wasn't mistaken! Probability is counterintuitive and evades even the brightest minds:
"Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem."
Barry Pasternack, Ph.D. California Faculty Association
"You're in error, but Albert Einstein earned a dearer place in the hearts of people after he admitted his errors."
Frank Rose, Ph.D. University of Michigan
"I have been a faithful reader of your column, and I have not, until now, had any reason to doubt you. However, in this matter (for which I do have expertise), your answer is clearly at odds with the truth."
James Rauff, Ph.D. Millikin University
"May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?"
Charles Reid, Ph.D. University of Florida
"I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns."
W. Robert Smith, Ph.D. Georgia State University
"You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?"
E. Ray Bobo, Ph.D. Georgetown University
"I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake."
Kent Ford Dickinson State University
"Maybe women look at math problems differently than men."
Don Edwards Sunriver, Oregon
"You are the goat!"
Glenn Calkins Western State College
"You made a mistake, but look at the positive side. If all those Ph.D.'s were wrong, the country would be in some very serious trouble."
Everett Harman, Ph.D. U.S. Army Research Institute
"Since you seem to enjoy coming straight to the point, I'll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and in the future being more careful."
Robert Sachs, Ph.D. George Mason University
"You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I'll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame!"
Scott Smith, Ph.D. University of Florida

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