Consider the following gambling game for two players, Black and white. Black puts black balls and white puts w white balls in a box. Black and white take turns at drawing at random from the box, with replacement between draws until either Black wins by drawing a black ball or white wins by drawing a white ball. Suppose black gets to draw first.

a) Calculate P(Black wins) and P(White wins) in terms of P = b/(b + w).

b) What value of p would make the game fair (equal chances of winning)?

c) Is the game ever fair?

d) What is the least total number of balls in the game, (b + w), such that neither player has more than a 51% chance of winning?