(Two geeks are waiting for the elevator on the seventh floor of a seven-story building. The elevator ascends to five and then begins to descend.)
Geek 1: I hate when people do that.
Geek 2: Yeah. Why should they care where the elevator’s going when they’re getting out anyway?
Geek 1: If they have to press a button, they might choose four at least. That would make sense.
Geek 2: Not so fast. If we disregard the intra-building trips — which are probably negligible in a residential building — then half of the elevator trips will originate from the first floor and the other half will start at floors two through seven, randomly distributed.
Geek 1: OK, I see where you’re going with this…
Geek 2: So, if we take a sample of twelve trips according to the expected distribution, six will originate from the first floor, and one each from floors 2 through 7, which comes to [sums 2 through 7 using Gauss's trick] 27 + 6 is 33, divided by 12, so in fact the optimal floor for the elevator to rest at, to minimize its travel time, is less than 3.
Geek 1: Immaterial, since floors are integral. But wait a second. Who on the second floor is going to use the elevator anyway?
Geek 2: Good point. So if we remove the second floor then we’ll add the floors for ten random trips, 5 + 25, divided by 10, looks like the third floor even.
Geek 1: No way it could be less than the third floor.
Geek 2: You live on seven. That sounds like special pleading.
Geek 1: Anyway the whole calculation is off. People tend to leave in the morning and return at night, therefore the optimal floor for the elevator is time-dependent…
Yes, it’s very, very sad.