An urn contains a white and b black balls. After a ball is drawn, it is returned to the urn if it is white; but if it is black, it is replaced by a white ball from another urn. Let Mn denote the expected number of white balls in the urn after the foregoing operation has been repeated n times.
(a) Derive the recursive equation
Mn+1 = (1 – 1/a + b) Mn + 1
(b) Use part (a) to prove that
Mn = a + b – b (1 – 1/a + b)n
(c) What is the probability that the (n + 1)st ball drawn is white?