A score that is neither highest nor lowest

Question

Four candidates each receive an independent score from the same continuous distribution, so no two scores tie.

What is the probability that one particular candidate ends up with neither the highest nor the lowest score (that is, ranked 2nd or 3rd of the four)?

Accepted Answer

By symmetry, the particular candidate is equally likely to hold each of the four ranks, so every rank has probability \frac{1}{4}. "Neither highest nor lowest" means rank 2 or rank 3, which is two of those four equally likely ranks: \frac{2}{4} = \frac{1}{2} = 0.5. Equivalently, the two "extreme" ranks (top and bottom) together carry probability \frac{2}{4}, and the rest is the middle. Takeaway. For n exchangeable draws, a specific one avoids both extremes with probability \frac{n-2}{n}.

A score that is neither highest nor lowest

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