Expected length of shortest interval containing numbers drawn at random

Question

A random idea:

If you draw $n$ numbers uniformly at random from $[0,1]$, what is the expected length $L_n$ of the shortest interval that contains all but one of them?

Clearly, we have

$$L_2 = 0$$

and

$$\lim_{n\rightarrow\infty} L_n = 1,$$

but even calculating $L_3$ is giving me troubles already.

Answer 1

I wonder if this would work:

Find the expected size of the smallest one $x_1$.

Find the expected size of the second to smallest $x_2$.

Find the expected size of the second to largest - by symmetry should be $1-x_2$.

Find the expected size of the largest - by symmetry should be $1-x_1$.

Then your answer should be $1-x_1-x_2$