Imagine a bank of three elevators along a wall. The elevators are in a straight line but they are not evenly spaced. Where do you stand in order to mi

Where to wait for an elevator

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2022-07-06 17:30:11

Imagine a bank of three elevators along a wall. The elevators are in a straight line but they are not evenly spaced. Where do you stand in order to minimize the expected distance you’ll need to walk to catch the first elevator that arrives?

If asked where they would stand and wait for the next of three elevators, unequally spaced along a wall, many students would choose to stand at the mean position. They think that by doing so they are minimizing the average distance to the elevator. They do not recognize that standing at the mean minimizes the average squared distance and that the minimal average distance to the elevator is in fact achieved by standing at the median.

Suppose you start out standing in front of the second elevator. If you move one foot to the left, you decrease the distance to first elevator by a foot, but you increase the distance to the other two elevators by one foot as each, so your average distance increases. Similarly, if you move one foot to the right, you decrease your distance to the third elevator but increase your distance to the other two so that you increase the average distance. Since you can’t move without increasing the average distance, you must have started at the best spot.

So standing in front of the second elevator minimizes the expected distance to the next elevator, assuming all three elevators are equally likely to arrive next.