How many floors does the elevator stop on?

Question

An elevator leaves the lobby with 5 riders. There are 3 upper floors, and each rider independently chooses one of the 3 floors uniformly at random. The elevator stops at a floor if at least one rider wants it.

On average, how many distinct floors does the elevator stop at?

Accepted Answer

Rather than track how the riders group up, place one indicator on each floor. Fix a floor. The elevator skips it only if none of the 5 riders chose it. Each rider avoids a given floor with probability 	frac23, so P(	ext{a given floor is skipped}) = \left(	frac23ight)^5 = \frac{32}{243}. Then a given floor gets a stop with probability 1 - 	frac{32}{243} = 	frac{211}{243}. Let Xi = 1 if floor i is visited; E[Xi] = 	frac{211}{243}. With 3 floors, E[X] = 3 	imes \frac{211}{243} = \frac{211}{81} \approx 2.605. Takeaway. "How many distinct groups appear" is an occupancy question: one indic…

How many floors does the elevator stop on?

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