Another problem that we will solve analytically later is the problem of finding the expected time that it takes a Markov chain to reach a state starting from another state. Consider the Markov chain with transition matrix and diagram below.

Build a command to let you approximate the average number of transitions necessary to reach state 9 , starting from a given one of the other states. Run your command for each of the initial states 6,7 , and 8 .

matrix8 = {{.375, .625, 0, 0, 0, 0, 0, 0, 0}, {.375, 0, 625, 0, 0, 0, 0, 0, 0}, {0, .375, 0, .625, 0, 0, 0, 0, 0}, {0, 0, .375, 0, .625, 0, 0, 0, 0}, {0, 0, 0, .375, 0, .625, 0, 0, 0}, {0, 0, 0, 0, .375, 0, .625, 0, 0}, {0, 0, 0, 0, 0, .375, 0, .625, 0}, {0, 0, 0, 0, 0, 0, .375, 0, .625}, {0, 0, 0, 0, 0, 0, 0, 0, 1}};