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, 1 Pli red, ni non-red] i=0,...,n Then balls are then randomly removed one at a time. Let Y denote the number of red balls in the first k selections, k = 1,..., n.

(a) Find P(Y, j, j = 0,..., n.

(b) Find P(Y-1=j),j=0,..., n.

(c) What do you think is the value of P(Y 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.