Thursday, April 10, 2008

Best explanation of the Monty Hall problem (Bayes)

John Tierney has been playing with explanations of the "Monty Hall" (Bayes theorem) problem for 17 years. That might be why he's provided the most succinct explanation I've come across ... (note: Monty knows where the car is, he can't open the door you picked, and he won't open the door for the car. That's important -- his actions provide new information. He's not picking randomly.)
Cognitive Dissonance in Monkeys - The Monty Hall Problem - New York Times

...Here’s how Monty’s deal works, in the math problem, anyway. (On the real show it was a bit messier.) He shows you three closed doors, with a car behind one and a goat behind each of the others. If you open the one with the car, you win it. You start by picking a door, but before it’s opened Monty will always open another door to reveal a goat. Then he’ll let you open either remaining door.

Suppose you start by picking Door 1, and Monty opens Door 3 to reveal a goat. Now what should you do? Stick with Door 1 or switch to Door 2?...

...You should switch doors.

... 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...
Probability problems are often asymmetric, they can be hard to solve in terms of the "correct choice", but easy to understand when considered when re-expressed in terms of the "wrong choice" (or vice-versa). That's what we see here.

Tierney's paragraph is a great example of expressing simple algebra in sentence form, but the key thing to recall is that Monty is adding new information because he doesn't choose randomly.

I'm fascinated by Bayesian probability. The mathematics is very simple, yet it can be very challenging to map correctly to the physical universe. On the other hand even a trivial understanding would greatly improve government and law enforcement. What a marvel!

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