In a random row, how many are taller than their neighbors?

Question

Ten people, all of different heights, stand in a single row in a completely random order. Call a person a "peak" if they are taller than every person standing directly next to them. The two people at the ends of the row each have just one neighbor.

On average, how many peaks are there in the row of 10?

Accepted Answer

Put one indicator on each position, but be careful: the ends have one neighbor while the interior positions have two. Interior positions (there are 10 - 2 = 8 of them): such a person has 2 neighbors. Among the 3 distinct heights of the person and both neighbors, the person is tallest in 	frac13 of orderings, so the peak probability is 	frac13. End positions (there are 2 of them): such a person has just 1 neighbor. The person is taller than that single neighbor in 	frac12 of orderings, so the peak probability is 	frac12. By linearity, adding the indicator expectations, E[	ext{peaks}] = 2 …

In a random row, how many are taller than their neighbors?

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