Around a round table, how many beat both neighbors?

Question

Twelve people, all with distinct scores, are seated in a completely random order around a circular table of 12 seats. Call a person a "local leader" if their score is higher than both of their two immediate neighbors.

On average, how many local leaders are there around the table?

Accepted Answer

Trying to reason about which people are leaders all at once is hard, because a leader forbids their neighbors from being leaders. Put one indicator on each seat instead. Fix a person. Around a circle, that person has exactly 2 neighbors. Look only at the 3 scores in play: the person and their 2 neighbors. All orderings of these 3 distinct scores are equally likely, and the person is a local leader exactly when their own score is the largest of the 3. That happens in 	frac13 of the orderings: P(	ext{a given person is a local leader}) = \frac13. Let Xi = 1 if person i is a local leader; E[Xi]…

Around a round table, how many beat both neighbors?

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