Mean minimum distance for N random points on a one-dimensional line

This can answered without any complicated maths.

It can be related to the following: Imagine you have $N$ marked cards in a pack of $m$ cards and shuffle them randomly. What is the probability that they are all at least distance $d$ apart? Consider dealing the cards out, one by one, from the top of the pack. Every time you deal a marked card from the top of the deck, you then deal $d$ cards from the bottom (or just deal out the remainder if there's less than $d$ of them). Once all the cards are dealt out, they are still completely random. The dealt out cards will have distance at least d between all the marked cards if (and only if) none of the marked cards were originally in the bottom $(N-1)d$. The probability that the marked cards are all distance d apart is the same as the probability that none are in the bottom $(N-1)d$.

The points uniformly distributed on a line segment is just the same (considering the limit as $m$$\rightarrow∞$). The probability that they are all at least a distance $d$ apart is the same as the probability that none are in the left section of length $(N-1)d$. This has probability $(1-\frac{(N-1)d}{L})^N$.

Integrating over $0$$\le$$d$$\le$$\frac{L}{(N-1)}$ gives the expected minimum distance of $\frac{L}{(N^2-1)}$.