Monty Hall problem -- solution and significance

 First of all, I should state for the record that Philadelphia jokes are actually neither true nor funny. Phillie is a great city to visit. It has a world famous art museum -- with a nearby Rodin museum -- world class symphony, sites of historical importance, famous and not-so-famous great restaurants, a vibrant independent theater and music scene.

Anyway, Marilyn's solution to the Monty Hall problem does depend on a couple of assumptions, which I'm not sure she spelled out explicitly. Maybe if she had it would have given all those outraged Ph.D.s a hint. The assumptions are that the rules of the game require Monty to show you a losing door, and to give you a choice to switch. This was not actually the case, which complicates matters, but for the sake of the puzzle the assumptions do pertain.

 

Here's how it works. When you first picked a door, you indeed had a 1/3 chance of getting the Corvette. That means there was a 2/3 chance it was behind one of the other doors. But when Monty shows you the losing door, there is now a 2/3 chance it's behind the other door you didn't pick. In other words, you now have more information, and you would be wise to revise your probability estimate accordingly.

There are two reasons this is interesting. One is that people generally tend not to revise their opinions when they get new information. We are inclined to dig in our heels and keep believing what we already believe no matter what new evidence comes our way. The other reason is that the pervasive approach to statistics in the sciences is what's called frequentism and it doesn't take into account any information other than the experimental or observational sample, which is why it was so hard for all those Big Professors to see how they should be incorporating additional information. I suppose there is a third reason, which is that peoples' intuitions about probability are largely wrong, and that evidently includes Big Professors who should know better. I'll have more to say about all this.