A Simple Computer Game Using a Markov Chain Starts at the Cave

Essay

A simple computer game using a Markov chain starts at the Cave or the Castle with equal probability and ends with Death (you lose) or Treasure (you win).The outcome depends entirely on luck.The transition probabilities are: $\begin{array}{c}\text { Next State }\\\begin{array}{lllll}&&\text { Cave }&\text {Castle }&\text {Death }&\text {Treasure }\\&\text { Cave } & .4 & .3 & .2 & .1 \\\text { Current }&\text { Castle } & .5 & .3 & .1 & .1 \\\text { State }&\text { Death } & 0 & 0 & 1 & 0 \\&\text { Treasure } & 0 & 0 & 0 & 1\end{array}\end{array}$


A.What is the average number of times you would expect to visit the Cave and the Castle, depending on which state is the starting state?
B.What is the mean time until absorption for the Cave and the Castle?
C.What is the likelihood of winning the game?