Question one
Mark Smith now works on the 13th floor of a 15 floor office building in Central Europe. 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 5pm, when he wants to go home, the elevator almost always goes up when it stops on his floor. What is the explanation?
Question two
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.
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 Mark arrives is negligible. This gives the probability 2/14 = 1/7 ≈ 0.1429 ≈ 14% that it is above floor 13 (which is when it will go down when it goes by this floor) when Mark wants to go home.
Question two
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 ≈ 25%.)
Note that this formula works for part 1 when n = 1.
Did you find the solution difficult or unusual?
This kind of result is quite normal for practical problem solvers who use maths to help businesses. They do not always have the problem statement clearly defined. To start to develop a solution to a real-world business problem they have to make certain assumptions. These assumptions are then updated as the results are analysed.
With thanks to the Tesco commercial science team.
