Katie has 8 black marbles, 12 clear marbles, and 10 blue marbles in a bag. if she picks one at random, what is the probability that she will pick a black marble? (enter your answer as a fraction in its reduced form.)

Let x have probability generating function gx (s) and let un generating function u(s) of the sequence uo, u1, satisfies p(x > n). show that the (1- s)u(s) = 1 - gx(s), whenever the series defining these generating functions converge.