Each year a tree of a particular type flowers once, and the probability that it has n flowers is (1 p)p n , n = 0, 1, 2, . . . , where 0 < p < 1. Each flower has probability 1 2 of producing a ripe fruit, independently of all other flowers. Find the probability that in a given year (a) the tree produces r ripe fruits, (b) the tree had n flowers, given that it produces r ripe fruits.