An interesting probability problem

Assume a particle moves forward with a step length i.i.d drawn from a uniform distribution on the interval (0, 1). Show the expected count of steps needed to escape from (0, 1).

Suppose at the nth step, the particle is Yn from its original position. And S steps is needed to escape from (0, 1). Then

E[S] = Pr[S>=1] + Pr[S>=2]+... = Pr[Y0<1]+Pr[Y1<1]+...=1+1/1!+1/2!+...=e

Aha! The expectation of the steps needed to escape from (0, 1) with a step length drawn i.i.d uniformly from (0, 1) is just e! The base of natural logarithm.