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,

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.

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1 P(i red, ni non-red): i=0,...,n n+1'