instruere...inlustrare...delectare Disputations

Friday, March 26, 2010

Game Show Theory

The Monty Hall Problem* can be stated this way:
There are three doors, numbered 1, 2, and 3, but otherwise identical. Behind one door is a prize; behind the other two doors is nothing. Each door is equally likely to have the prize behind it.**

Is the probability of choosing the door with the prize behind it higher if you choose one door or if you choose two doors?
It can be stated this way, but it isn't, because as always mathematical accuracy sucks the fun out of everything. The fun in this case is watching people struggle to understand the counterintuitive implications of a door with no prize behind it being opened.***

I think what makes the implications counterintuitive is the order in which the game unfolds. If it's laid out all at once, as I did above, the answer is obvious.

When it comes at us in stages, though, we get confused. We know at least one of the unchosen doors has nothing behind it, so we assume subsequently learning a particular door has nothing behind it tells us nothing new. And if we're told nothing new, then both the door we chose and the other remaining door each still has a 1/3 probability of having the prize, so why change? And if 1/3 + 1/3 doesn't add up to 1, we just normalize the probability vector to <0.5, 0.5>.

Ha! No, we stick with the door we chose because it's the door we chose, and no fork-tongued game show host is going to talk us out of the prize we earned by our own gloriously autonomous act of free will.

Or perhaps, as good poker players, we recalculate the pot odds and figure there's no angle in changing our mind now.

In any case, one way or another it comes down to this: we don't properly account for the fact that the game show host knows which door has the prize behind it. We are given new information, and we don't notice what that information means.

* Not to be confused with the Wayne Grady Problem, which is having a lot of talent but no suitable outlet (cf the Dick Van Dyke Problem).

** While I'm here, I can't be the only one who, as a child, thought it would be great to win a donkey.

*** For those who care about such things, I'd thought of bringing up the Monty Hall Problem in the context of the Influence of a Vote Problem, then decided against it forgot about it, then saw it mentioned in the comments on this post, then decided to write about it, then didn't have time, then saw it mentioned again, and finally decided to write about it for sure.