This is question regards on the statistics and mathematics concept. Please show me the step-by-step solution. 2. Consider the following game of chance. Initially, a player is given a single token. During each turn the player rolls a 6-sided dice. If the result is a 1 or 2 the player discards a token. If the result is a 3 or 4 the player picks up an additional token. If the result is a 5 or 6 the player picks up an additional two tokens. The game ends in a win if the player discards all tokens and a loss if the player accumulates four or more tokens. At any given time the player is in one of five states W, T, 27, 37, L. W: Player has 0 tokens and has won the game (absorbing state). T : Player has 1 token. 2T : Player has 2 tokens. 3T : Player has 3 tokens. L: Player has 4 or more tokens and has lost the game (absorbing state). (a) Draw a state transition diagram. (b) List all state sequences that lead to a win in exactly 6 turns. (c) Calculate the probability of each state sequence listed in part (b) and use the results to calculate the probability that a player wins the game in exactly 6 turns. [P(Sk=W) P(Sk = T) (d) Set Pk P(Sx = 2T). Find an initial probability vector po and a transition matrix A P(Sk = 3T P(Sk = L) such that for all k > 0 we have Pk+1 = Apk (e) calculate pe and ps. Use the results to verify your answer to part (c). (f) calculate P1000 (8) Use the numerical evidence from part (f) to conjecture the value of the following limits. Probability of Winning Game = lim P(Sk = W) = Probability of Losing Game lim P(Sk = L) =