The Elevator Paradox

What’s going on

You are on floor 9 of a 10-floor building and you want to go down. You press the button and watch the doors. When an elevator finally comes, it is going up. Next day the same thing happens, and the day after. It feels as if the building is against you.

The elevator is not being unfair. It goes up past your floor exactly as often as it goes down past it. Yet press Wait 1,000 times in the scene above: the first elevator to reach floor 9 is going up about 89% of the time.

Why

Follow one elevator that rides non-stop from floor 1 to floor 10 and back, at an even speed. Measure its trip in floors: up 9 floors, down 9 floors, 18 in all. Now look at it from floor 9:

• it passes you going up, goes one floor to the top, comes one floor back, and passes you going down: that gap is only 2 floors long;
• then it goes all the way down to floor 1 and back up to you: that gap is 16 floors long.

You walk up to the elevator at a random moment. You are 8 times more likely to arrive during the long gap, because it is 8 times longer. And at the end of the long gap the elevator is coming up from below. So 16 out of 18, or 8 times in 9, the first elevator you see is going up.

Up and down passes happen equally often, but they are not evenly spaced. A random moment tends to fall in the long gaps, so the pass that ends a long gap is the one you usually see. The same thing makes the bus you wait for seem to come less often than the timetable says.

Press Test every floor. Near the top the first elevator is mostly going up; near the bottom it is mostly going down; in the middle it is half and half. With one elevator, on floor k of N, the chance that it is going up is (k − 1) ÷ (N − 1).

More elevators

Real buildings have several elevators. Choose 3 in the scene: on floor 9 the share going up drops to about 73.5%. If the elevators run independently, the first to arrive is whichever happens to be closest in time, and that can be one coming down from floor 10. The more elevators, the closer the answer gets to one half:

Elevators	1	2	3	4	6
Floor 9 of 10, first one going up	88.9%	80.2%	73.5%	68.3%	61.1%

The effect gets weaker but never quite disappears. (Real elevators stop at floors and answer calls, so in a real building the numbers are different again.)

Try it yourself

1. The middle floor

A building has 11 floors and one elevator. You wait on floor 6. How often is the first elevator going up?

Answer

Exactly half the time. Floor 6 is in the middle: there are 5 floors above and 5 below, so the two gaps between passes are equally long.

2. Near the bottom

Same 10-floor building with one elevator, but now you wait on floor 2 and want to go up. How often is the first elevator going your way?

Answer

Only one time in nine. Turn the building upside down and it is the floor-9 case: the long gap is now spent above you, so the first elevator usually comes down from above. The chance that it is going up is (2 − 1) ÷ (10 − 1) = 1/9, about 11%.

3. Two elevators

On floor 9 of 10, with two elevators running independently, the first to arrive is going up about 80.2% of the time. Show that the exact value is 65/81.

Answer

Write r = 7/9: it is the difference between the long gap and the short gap, 16 − 2, divided by the whole trip, 18. With m independent elevators the chance is ½ + ½ × rᵐ. For one elevator that gives ½ + ½ × 7/9 = 8/9. For two: ½ + ½ × 49/81 = 130/162 = 65/81 ≈ 80.2%.

Go deeper

• The puzzle comes from the physicists George Gamow and Marvin Stern, who wrote about it in their 1958 book Puzzle-Math. Gamow worked near the bottom of a building and found the first elevator was usually going down; Stern, near the top, found it was usually going up.
• How the effect fades with more elevators was worked out by Donald Knuth in a 1969 paper on the Gamow–Stern elevator problem.