The Drunkard’s Walk and the most counter-intuitive math problem
I don’t think there’s a more counter-intuitive math problem than the Monty Hall problem.
It’s based loosely on the TV show Let’s Make a Deal, but was made popular after a Parade magazine columnist was berated, by literally thousands of Ph.D.s and mathematicians, for her correct answer.
Here’s the problem:
Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which 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 make the switch?
The answer and reasoning for it is best described by Wikipedia here. I think the decision tree diagram explains it most clearly. The most mind-boggling aspect of the answer is that it’s a 2-to-1 advantage!
I completely forgot about this problem until I was reading The Drunkard’s Walk: How Randomness Rules Our Lives written by Leonard Mlodinow, who also co-authored A Briefer History of Time with Stephen Hawking. Mlodinow’s new book contains this and other “real-life” probability and statistics problems (How a mathematician won a small fortune by playing roulette; Use of the Prosecutor’s Fallacy in the O.J. Simpson trial; Why firing Merril Lynch’s CEO for losing on subprime investments doesn’t make sense) interspersed with a surprisingly interesting history of famous mathematicians.
I think it’s a fantastic book to read if you’re on a math kick, but a boring one to read if you’re having one of those days where you just don’t feel like calculating your waiter’s tip in your head.