28. Polya’s urn model supposes that an urn initially contains r red and b blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let Xk be the number of red balls drawn in the first k selections.

(a) Find E[X1].

(b) Find E[X2].

(c) Find E[X3].

(d) Conjecture the value of E[Xk], and then verify your conjecture by a conditioning argument.

(e) Give an intuitive proof for your conjecture.

Hint: Number the initial r red and b blue balls, so the urn contains one type i red ball, for each i = 1, . . . , r; as well as one type j blue ball, for each j = 1, . . . , b.

Now suppose that whenever a red ball is chosen it is returned along with m others of the same type, and similarly whenever a blue ball is chosen it is returned along with m others of the same type. Now, use a symmetry argument to determine the probability that any given selection is red.