Mark Smith works on the 13th ﬂoor of a 15 ﬂoor office building. There is only one elevator. The elevator moves continuously through ﬂoors one to 15 and then back down to one.
The elevator only stops on a ﬂoor on which the button has been pressed. It does not change direction unless it is at the top or the bottom of the building.
Assume that time spent loading and unloading passengers is very small compared to the travelling time.
Mark complains that at 5pm, when he wants to go home, the elevator is almost always going up when it stops on his ﬂoor. Is he correct? What is the explanation for this?
As there is only one elevator, 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 the elevator is above ﬂoor 13, so there is only a 14% chance that the elevator will be heading down when Mark wants to head home. Perhaps he should use the stairs.
With thanks to the Tesco commercial science team.