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.