Consider the independent trials each of which results in outcome i, i = 0,1,...,k with probability p_i , \sum_{i = 0}^{k} p_i = 1. Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome.

(a) Find P(N = n), n \geq 1.

This is the only one that is really giving me trouble. Here is what I have:

P(N = n) = \sum_{j = 0}^k P(N = n | X = j)P(X = j)
= \sum_{j = 0}^k P(N = n, X = j)
= \sum_{j = 0}^k P(N = j)...?

(b) Find P(X = j), j = 1,...,k.

P(X = j) = p_j

(c) Show that P(N = n, X = j) = P(N = n)P(X = j)

All I need is part (a) and this becomes trivial.

Any help? Thanks.