The Monty Hall Problem

Sometimes even I’m amazed at the things I find myself occupied with instead of sleeping. Things such as, The Monty Hall Problem. The Monty Hall Problem is arguably the most famous math/probability questions of recent times. And, I don’t think you have to be a probability or math nerd to appreciate it — I’m not, and I do.

Do you remember a tv game show called Let’s Make a Deal? Oh well, that’s ok. The idea of the game show sets up the problem thusly:

This really became a big deal when addressed by Marilyn vos Savant (famous for holding the “Guinness Book of World Records distinction for Highest IQ”) in Parade magazine’s “Ask Marilyn” column. Needless to say, she’s pretty smart. Her answer was: “Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance.”

A diagram helps to visualize it:

Now, think about that for a second. And then, read some of the flurry responses that this generated.  What’s so tantalizing about this, of course, is that it’s so goddamn counter intuitive until you really think about it and, if you carefully read through the explanation (still confused? an explanation really does help to make all of this much clearer). John Tierney, who also wrote about this at length back in 1991 (“Behind Monty Hall’s Doors: Puzzle, Debate and Answer?“), writes more recently:

“This answer goes against our intuition that, with two unopened doors left, the odds are 50-50 that the car is behind one of them. But when you stick with Door 1, you’ll win only if your original choice was correct, which happens only 1 in 3 times on average. If you switch, you’ll win whenever your original choice was wrong, which happens 2 out of 3 times.” 


More Monty Hall Problem-related links!

  1. The Monty Hall Problem is also treated at length in Leonard Mlodinow’s excellent and interesting book, Drunkard’s Walk: How Randomness Rules Our Lives.

  2. There is a lot to be said about this topic (enough to fill an entire book, The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser, Oxford University Press, 2008).

  3. The official Let’s Make a Deal website has a brief chronology of how that problem has appeared and reappeared in recent years.

  4. A close relative of this is also called The Three Prisoners Problem.

  5. And check out this cool interactive version from The New York Times: “Interactive Feature: The Monty Hall Problem” (click on the image below to play the game at “Play enough rounds and the best strategy will become clear: You should switch doors.”

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