14.9 [Polya’s urn] Suppose an urn contains one white ball and one green ball. At time 1, a ball is selected randomly and a ball of the same color is added to the urn so that there are now three balls in the urn. Then at time 2, a ball from the urn is selected randomly and a new ball of the same color is added. The process continues like this.

Note that after n selections are completed, the urn contains n + 2 balls. These n + 2 balls are made of Zn + 1 white balls and n − Zn + 1 green balls, where Wn is the number of new white balls that have been added to the urn. Similarly, n − Zn is the number of new green balls.

a) Show that P(Zn = i) = 1 n+1 for all i ∈ {0, 1,...,n}.

b) Show that the process