An urn contains a white balls and b black balls. After a ball is drawn, it is returned to the urn if it is white; but if itis black, it is replaced by a white ball from another urn. Let M(n) denote the expected number of white balls in the urn after the foregoing operations has been repeated n times.

(a) Derive the recursive equation

M(n+1) = 1+[1-1/(a + b)]M(n)

(b) Use part (a) to prove that

M(n )= a + b - [b (1-1/(a + b))^n]

(c) What is the probability that the (n + 1)st ball drawn is white?