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Peaks in a random line

Expected valueHard

Balls numbered $1$ to $10$ are placed in a row in a uniformly random order. A ball in one of positions $2$ to $9$ is a peak if its number is larger than the numbers of both neighbours. What is the expected number of peaks? Give a fraction a/b in lowest terms.

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Hint

For one interior position, look only at the relative order of the $3$ balls in positions i− $1$ , i and i+ $1$ .

Solution

Take an interior position and its $2$ neighbours. These $3$ balls are equally likely to be in any relative order.
The middle position is a peak exactly when the largest of the $3$ balls is in the middle, which has probability $\frac{1}{3}$ .
There are $8$ interior positions, from $2$ to $9$ .
By linearity the expectation is $8 \times \frac{1}{3} = \frac{8}{3}$ .

Answer: $\frac{8}{3}$

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