In the one-elevator case, we can reasonably assume that the elevator is equally likely to be
at any point between floor 1 and floor 15 at any point in time. We can also assume that the
probability that the elevator is exactly on the 13th floor when Smith arrives is negligible. This
gives the probability 2/14 = 1/7 0.1429 that it is above floor 13 (which is when it will go
down when it goes by this floor) when Smith wants to go home.
Let’s have n elevators now. Call the unbiased portion the part of the elevators route up from
floor 9 to the top and then down to floor 13. Any elevator at a random spot of the unbiased
portion is equally likely to go up or down when it goes by the 13th floor. Moreover, if there is at
least one elevator in the unbiased portion, all elevators out of it do not matter. However, if no
elevator is in the unbiased portion, then the first one to reach the 13th floor goes up. Therefore
the probability that the first elevator to stop at 13th floor goes down equals 1
2 (1 − (10/14)n).
(For n = 2 it equals approximately 0.2449.)
