Expected Number of Good Pairs This is a question I had in my interview: we have $N$ i.i.d Uniform$(0, 1)$ random variables.  Define a good neighbor for $x_i$ as the point that is closest to $x_i$ in absolute value. We call a pair $(x_i, x_j)$ a good pair if $x_i$ is $x_j$'s good neighbor and $x_j$ is $x_i$'s good neighbor. What's the expected number of good pairs?

The order statistics $x_{(1)}\le x_{(2)} \le \cdots \le x_{(N)},$ when suitably scaled, have iid exponential gaps $y_i = x_{(i+1)} - x_{(i)},$ and those gaps are the distances between neighboring points. A "good" gap is one that is smaller than the gaps immediately before and after it.
There are $2$ gaps  with only one adjacent gap (namely, $y_1$ and $y_{N-1}$).  The chance that such a gap is smaller than its neighbor is $1/2$ because the two values are iid (and continuously distributed).  Otherwise the chance a gap is smaller than two neighboring gaps is $1/3,$ for a comparable reason.  There are $N-3$ such gaps.  Therefore the expectation is
$$2\left(\frac{1}{2}\right) + (N-3)\left(\frac{1}{3}\right) = \frac{N}{3}.$$
This analysis provides an unexpected detail: except for the pairs of points at the ends of the interval (each of which has a $1/2$ chance of being good), every other adjacent pair has a $1/3$ chance to be good.
