Problem 31

Gamow, Stern, and Elevators (1958)

Problem. A person is waiting for the elevator on the ath floor in a building and wishes to go down. The building has b (> a) floors. Assume the elevator is continually going up and down all day long. Calculate the probability that when the elevator arrives, it will be going down.

Solution. The elevator can come down to the person from any one of the (b − a) floors above or can come up from any one of the (a − 1) floors below. Therefore, the probability of the elevator going down is

# 31.1 Discussion

This intriguing little problem was adduced by the physicists George Gamow (1904–1968) (Fig. 31.1) and Marvin Stern (1935–1974), and is eponymously called the Gamow–Stern Elevator Problem.^{1} The problem first appeared in their book Puzzle-Math (Gamow and Stern, 1958). As the story goes, Gamow had an office on the second floor of a seven-story building while Stern had one on the sixth floor. When Gamow wanted to visit Stern, he noticed that in about five times out of six the elevator stopping on his floor would be going downward. On the other hand, when Stern wanted to visit Gamow, in about five times out of six the elevator stopping on his floor would be going upward. This is somewhat counterintuitive because for Gamow the elevator was more likely to be coming down, yet for Stern the same elevator was more likely to be going up.