3. (15 points) Suppose an urn initially contains 4 blue balls and 1 red ball. Each time you randomly select a ball from the urn, you do the following: if you select a blue ball, you place it back in the urn, but if you select a red ball, you place it back in the urn, and you place another red ball in the urn. This continues indenitely. Each time you randomly select a ball from the urn, assume each ball is equally likely to be selected. (a) Let X\" denote the number of red balls found in the urn after having made 12 selections. Model {Xn}n20 as a DTMC, and describe both its state space S, and its generator 3. (b) Assume X0 = 1 (as is assumed at the beginning of the problem). Find IP(X3 = 2). (c) Suppose you continue selecting balls from the urn in this manner, until you either (a) select 8 blue balls, or (b) select 2 red balls. If you select 8 blue balls rst, you win, but if you select 2 red balls rst, you lose. Find the probability you win. Hint: you may want to modify your DTMC so that state 3 is an absorbing state. Think carefully about precisely what has to happen in order to select 8 blue balls before 2 red balls