Let U1,U2, . . . be a sequence of independent uniform (0, 1) random variables, and let N = min{n  2: Un > Un−1}

and M = min{n  1: U1 + · · · + Un > 1}

That is, N is the index of the first uniform random variable that is larger than its immediate predecessor, and M is the number of uniform random variables we need sum to exceed 1. Surprisingly, N and M have the same probability distribution, and their common mean is e!