[blah_blah]

the first problem is a fairly simple consequence of the fact that the simplex bounded by the coordinate planes x_0 = ... = x_n = 1 and x_1+...+x_n = 1 has n-dimensional measure equal to 1/n!, but imo a better solution technique is as follows. Let E(t) denote the expected number of turns to reach t starting at 0. We have, for 0<t\leq 1,

E(t) = (1-t) + \int^t_0 [E(x)+1]dx

This is an integral equation with solution E(t) = a\exp(t). Note that as t decreases to 0, E(t) tends to 1. In particular we conclude that a=0 and E(t) = exp(t), so the desired answer is E(1) = \exp(1) = e