Question: Conditional expectations in polya's urn scheme. An urn contains 1 black and 2 white balls. One ball is drawn at random and its noted. The ball is replaced in the um, together with an additional ball of its color. There are now four balls in the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.
a) Let Bn be the number of black balls in the urn just before the nth ball is drawn, (Thus B1 is 1.) For n > 1, find E(Bn+1 |Bn).
b) For n > 1, find E(Bn). [Hint: E(B1) = 1: now use part a) and induction on n.]
c) Fond n > 1, what is the expected proportion of black balls in the urn just before the nth ball is drawn?