Is the Elevator Puzzle a Math Paradox or a Paranoid Delusion?

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Resolving the Simplified Elevator Puzzle

Consider this a simple probability exercise. At any moment, the most recent “status” of each elevator is a floor and a direction. For example, “On the first floor and about to go up” is the state “1 : Up”. There are 18 such states, as shown in the first image.

These states are {(1 : up), (2 : up),…,(9 : up), (10 : down), (9 : down), (8 : down),…,(3 : down), (2 : down)}.

When the elevator first opens on your floor, if it is going in the direction you want, you will be “Happy!” and board the elevator. If it is going the other way, you will be “Annoyed!” by the extra waiting time.

At the moment when you step into the second-floor elevator lobby, the elevator is equally likely to be in any one of these 18 states.

But only under those 2 conditions, out of 18, will you observe the elevator first open its door on the second floor while heading up. In the other 16 cases, you will first see the elevator open its door before going down to the first floor. Otherwise, after a variable delay time, you will see the elevator open its door while headed in the “wrong” direction.

The Ninth Floor is No Better

The probabilities are the same when you step into the lobby on the ninth floor. The elevator may have been on the tenth, or is just coming down from there; those are the 2 fortunate conditions out of 18.

Summary of the Simplified Elevator Puzzle

Regardless of whether you go from 2 to 9 or vice versa, in this ten-floor building with one relentless elevator, you only have 2 chances in 18 (or 0.111…) that the elevator will be going in the direction you want when it first opens its door.

In other words, you observe that you only have an 11% chance that the elevator will head in the right direction the first time you see it arrive. But it should be going up half the time, and down half the time.

The Observer Versus the Observation

In fact half of the times when it passes your floor it is going up.  If you were to stand quietly in the second-floor lobby for a full day and keep a log of the elevator’s activity, you observe it actually does have a 50% chance of being in state “2 : up” and “2 : down”. These are the only states you can observe from the second-floor lobby.

Because you normally take the elevator as soon as it is in the correct state, you do not take a proper statistical sampling of the elevator’s activity. Your 11% success ratio is due to the combination of where the elevator spends its time compared to your floor, and mainly because you cut short the “experiment” once your needs are met.

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