Mr. Smith works on the 13th ﬂoor of a 15 ﬂoor building. The only elevator moves continuously through ﬂoors 1, 2, . . . , 15, 14, . . . , 2, 1, 2, . . . , except that it stops on a ﬂoor on which the button has been pressed. Assume that time spent loading and unloading passengers is very small compared to the travelling time.

Mr. Smith complains that at 5 pm, when he wants to go home, the elevator almost always goes up when it stops on his ﬂoor. What is the explanation?

Now assume that the building has n elevators, which move independently. Compute the proportion of time the ﬁrst elevator on Mr. Smith’s ﬂoor moves up.

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