16 December 2007

The elevator paradox

Saw Friday's episode of Numb3rs, which I usually skip because the mathematical content is usually laughable from my point of view. But I learned about the elevator paradox, which says that when you're at the bottom of a building, usually the first elevator which comes will be going down, but when you're at the top of a building, it'll usually be going up. The reason, according to the Wikipedia article, is:
Essentially, the explanation seems to be this: a single elevator spends most of its time in the larger section of the building, and thus is more likely to approach from that direction when the prospective elevator user arrives. An observer who remains by the elevator doors for hours or days, observing every elevator arrival, rather than only observing the first elevator to arrive, would note an equal number of elevators traveling in each direction.

But this raises an interesting question: how do we know that you'd have an equal number of elevators traveling in each direction? This is, it appears, basically by conservation of elevators. If in general more elevators go upwards than downwards past a certain point, then the number of elevators below that point will steadily decrease. Surprisingly, none of the online expositions seem to mention this. They seem to think it's obvious that the first elevator passing a point has an equal probability of going up and down. But I've spent enough time with probability to know it's ridiculously counterintuitive.

Also, should Wikipedia have mathematical proofs?, from Slashdot. I haven't thought about this; if I think of anything worth saying I'll post it here, but the link alone didn't seem worth its own post.

1 comment:

John Armstrong said...

Quiz question: how does this relate to the sophomore jinx?