Thursday, September 18, 2008

This post is of no interest to you unless you have trouble with the Monty hall problem.

I never believed the "correct" answer to the "Monty Hall problem." (In brief: you're on a game show and pick a door, hoping for a new car. The host opens one of the other doors and reveals a goat, and then gives you the option of switching to the unopened door you didn't pick. Should you do it? The problem presumes you prefer cars to goats, despite goats being cuter and more ecologically friendly and producers of feta.) It turns out that you always should, even though in my mind it shouldn't matter. No matter how many times I heard it explained I still couldn't get it. In part this is because if the host opened one of the other doors without knowing what lay behind it, and just happened to encounter a goat, the probabilities would in fact be equal. But I finally read an explanation which helps me to wrap my head around it:

It may be easier to appreciate the solution by considering the same problem with 1,000,000 doors instead of just three (vos Savant 1990). In this case there are 999,999 doors with goats behind them and one door with a prize. The player picks a door. The game host then opens 999,998 of the other doors revealing 999,998 goats—imagine the host starting with the first door and going down a line of 1,000,000 doors, opening each one, skipping over only the player's door and one other door. The host then offers the player the chance to switch to the only other unopened door. On average, in 999,999 out of 1,000,000 times the other door will contain the prize, as 999,999 out of 1,000,000 times the player first picked a door with a goat. A rational player should switch.

Thanks Wikipedia!


Robin said...

I worked this for ages in college before it sank in. I've found different ways of phrasing the idea to be more helpful for different people, but probably my most successful is this: your odds reverse. If you were wrong before, you're right when you switch, and vice versa.

My brother and I, earlier, actually dummied up a set of doors to watch it happening for a while and keeping a tally, which is what made me believe it even though I didn't really get it at that point. Now I kind of can't remember what seemed so weird and counterintuitive about it before.

Mala said...

I actually get it now. And I am a Let's Make a Deal fan so I will be picking the 1975 Chevy every time now instead of the donkey. Although I do agree with you I usually would prefer the donkey to the prize. I wonder what they would say if you actually wanted to take it home.

Nana said...

I need a diorama.