All the news that's fit to calculate (and brought to you by The Clash)

April 14, 2008 in Math

Obviously the most exciting thing that happened last week was the publication of a popular article all about statistics. Yes, listeners (readers?), the NYTimes reviewed a possible flaw in a well-known psychology experiment based on the mathematics of a well-known game show prize.

Of course I'm referring to the Monty Hall problem, which used to be an interesting riddle (in fact, it used to be an interesting game show) and now has become more of a novelty. Apparently, however, solving it remains the defining criteria to join an elite blackjack squad.

The problem goes like this: Monty Hall shows you three doors, one of which has a car behind it. The other two hide goats. All you have to do is pick the right door. The trick is that after you make a selection, Monty opens a door you did not choose and reveals one of the goats. He then gives you the option to stay with the door you originally chose or switch to the (sole) other door. So the question is, Should You Stay or Should You Go?

The intuitive answer is that it doesn't matter - there are two doors, one with a car and one with a goat, so it should be 50-50. But since I said that's intuitive, you know it must be wrong. In fact, it turns out you should always switch to the other door.

When you made your original selection, you had a 1/3 chance of being correct. When Monty eliminates a door, you STILL only have a 1/3 chance of being right. That's the tricky realization: because you choose before the elimination, the elimination does not affect your probability of being correct. However if you change your choice, then you can take advantage of the fact that there was a 2/3 chance your original pick was wrong. Earlier, you couldn't make use of that 2/3 probability because it was divided evenly over two other doors. Now, there is only one "other door," so the 2/3 probability that a door other than yours is correct has been concentrated in one place. Switch, and you double your chances of winning.

And so The Clash was quite correct:

If I go there will be trouble / And if I stay there will be double.

(I can only assume they were referring to the probability of picking a goat.)

Honorary mention goes to the Numbers Guy at the WSJ, who ran an actual probability quiz in his blog last week.

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