65. The number of red balls in an urn that contains n balls is a random variable that is equally likely to be any of the values 0, 1,...,n. That is, P{i red,n − i non-red} =

1 n + 1

, i = 0,...,n The n balls are then randomly removed one at a time. Let Yk denote the number of red balls in the first k selections, k = 1,...,n.

(a) Find P{Yn = j },j = 0,...,n.

(b) Find P{Yn−1 = j }, j = 0,...,n.

(c) What do you think is the value of P{Yk = j },j = 0,...,n?

(d) Verify your answer to part

(c) by a backwards induction argument. That is, check that your answer is correct when k = n, and then show that whenever it is true for k it is also true for k − 1,k = 1,...,n.