3. In a box, Lynn has b batteries of which d are dead. She tests them randomly and one by one. Every time that a good battery is drawn, she will return it to the box; every time that a dead battery is drawn, she will replace it by a good one.

(a) Determine the expected value of the number of good batteries in the box after n of them are checked.

(b) Determine the probability that on the nth draw Lynn draws a good battery. Hint: Let Xn be the number of good batteries in the box after n of them are checked. Show that E(Xn | Xn−1) = 1 + * 1 − 1 b , Xn−1. Then, by computing the expected value of this random variable, find a recursive relation between E(Xn) and E(Xn−1). Use this relation and induction to prove that E(Xn) = b − d * 1 − 1 b ,n . Note that n should approach ∞ to get E(Xn) =

b. For part (b), let En be the event that on the nth draw she gets a good battery. By conditioning on Xn−1 prove that P (En) = E(Xn−1)/b.