a markov probability question

Sep 2019
6
0
New york
Three friends are playing paintball. Maria is a good shot, hitting her target half of the time; Fan isn’t bad, hitting her target one third of the time; Peter is new to the game, and only hits his target one sixth of the time. Each round, all three players take a single shot at the same time, and anyone who is hit leaves the game. When all three people are still in the game, all players target the best shot out of their opponents; that is, Fan and Peter both target Maria, while Maria targets Fan. Write out a Markov chain model of the game, including a transition diagram and the matrix of transition probabilities. What is the probability that Peter is still in the game after three rounds of play?

this was given to me in a linear algebra class and it is the only homework question I didn't quite know how to approach... I managed to draw a diagram but I don't really know how to put this into a matrix? I know for the first round ( or row) it would be =

t1=1/2F 0P 1/2M and I assume the following rows would be T+1 and T+2 but how do you computer the probabilities? Thanks!
 
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