TITLE: What is the probability that the range of a set of N randomly chosen real numbers in [0, 1] is less than the reciprocal of N?
QUESTION [6 upvotes]: (Random number with uniform distribution over [0, 1])
For clarification, in the case where N = 2, we can use geometric probability. On the coordinate plane consider points with 0<=x,y<=1. The condition is satisfied on a diagonal band of area 3/4 from the origin to (1,1). Similarly, with N = 3 the volume of the space in which the condition is satisfied is 7/27.

REPLY [14 votes]: The keyword is the order statistics. The distributions of the maximum and minimum values of a sample of $n$ independent uniformly distributed random variables are given respectively by the laws
$$U_{max}\sim \mbox{Beta}(n,1),\qquad U_{min}\sim \mbox{Beta}(1,n).$$ 
The range $U_{max}-U_{min}$ has a $\mbox{Beta}(n-1,2)$ distribution (see, e.g., Section 2.5 of A First Course in Order Statistics) so
$$\mathbb P\{U_{max}-U_{min} < a\}=\frac{1}{B(n-1,2)}\int_{0}^{a}x^{n-2}(1-x)dx=na^{n-1}-(n-1)a^n.$$