Random walk in one dimension. A person (say, in an unstable state of mind/body) is moving in one dimension, with coordinate x, starting at x = 0. Assume: i.) that s/he moves in steps of length l, ii.) that the probability that s/he takes a step to the left is p, while the probability of taking a step to the right is q = 1 − p and iii.) that all the steps are independent (i.e. the probability of taking the n + 1-th step left or right is independent on what the previous n steps were). One of the questions ask: Find the average number of steps to the right, <nR>, taken after N steps. This is what I got: <nR> = sum i=0toN i*(N choose i) * p^(i) * (1-p)^(N-i) A played around with it but i cant seem to get it into a nicer form. Other questions then ask for the variance and to compare it to the mean, so im sure i have to somehow eliminate the summation sign. There's a hint saying to use the fact that p d/dp(p^n) = np^n. Is there a way to simplify this?