The Monty Hall Problem

Let's start with a very standard probability question (the Monty Hall problem) that was put to Marilyn vos Savant.

Q: 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?

A: 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 #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 #777,777. You'd switch to that door pretty fast, wouldn't you?

Now, I understand that the immediate instinct is to think that door #1 and door #2 each have a 50-50 chance of having the car. However, if you think about it for a few minutes (and I would certainly expect you to do so before writing Ms. vos Savant to tell her she's wrong), you should pretty quickly realize that the host will always open the door from 2,3 that does NOT have a car, and therefore the probability of the remaining door is 2/3 as opposed to 1/3 for the original door #1. I would certainly expect a PhD in math to come to this conclusion in short order. Even if they couldn't do it on their own, this is a standard problem covered in dozens of introductory statistics textbooks (hell, it's so well understood it's got a freakin' name), so I would at least expect a PhD in math to LOOK IT UP! However, I guess my expectations would be sadly shattered, not once but nine times (the names have been bolded to call out the guilty)...

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

DUH, we're not talking about quantum tunnelling.

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

And you have a 0% chance of being correct with this reasoning. I didn't know so much of the mathematical illiteracy in this country resided in mathematics faculties.

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

Preferably not the one that he uses to teach.

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

And with future lectures at GSU.

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

Apparently the necessary number of IRATE mathematicians is n where n>9. To get her to change her mind, she needed to hear from exactly 1 CORRECT mathematician.

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

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

Serious trouble indeed! What a sad state of affairs that not only can't these nine PhD's not get the correct answer on their own, but they can't even recognize the correct answer when it is put before them in excruciating detail. I am so sad for their students, more so than for the country!

[Thanks to WM for the link]