37. An urn contains ???? white and ???? 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 ????ₙ denote the expected number of white balls in the urn after the foregoing operation has been repeated ???? times.

(a) Derive the recursive equation

????ₙ₊₁ = (1 − ????/(???? + ????))????ₙ + 1

(b) Use part

(a) to prove that

????ₙ = ???? + ???? − ????(1 − ????/(???? + ????))ⁿ

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