Six boys (Dick, Harry, Joe, Mark, Sam and Tom) play catch. If Dick has the ball,

Question

Six boys (Dick, Harry, Joe, Mark, Sam and Tom) play catch. If Dick has the ball, he is equally likely to throw it to Harry, Mark, Sam and Tom. If Harry gets the ball, he is equally likely to throw it to Dick, Joe, Sam and Tom. If Sam has the ball, he is equally likely to throw it to Dick, Harry, Mark and Tom. If either Joe or Tom gets the ball, they keep throwing it to each other. If Mark gets the ball, he runs away with it. Obtain the transition probability matrix and classify the states of the chain.

Explanation / Answer

Dick Harry Joe Mark Sam Tom Dick 0 0.25 0 0.25 0.25 0.25 Harry 0.25 0 0.25 0 0.25 0.25 Joe 0 0 0 0 0 1 Mark 0 0 0 1 0 0 Sam 0.25 0.25 0 0.25 0 0.25 Tom 0 0 1 0 0 0 The following are the states of the chain [D, H, J, M, S, T]* Dick Harry Joe Mark Sam Tom Dick 0 0.25 0 0.25 0.25 0.25 Harry 0.25 0 0.25 0 0.25 0.25 Joe 0 0 0 0 0 1 Mark 0 0 0 1 0 0 Sam 0.25 0.25 0 0.25 0 0.25 Tom 0 0 1 0 0 0 0.25H+0.25S=D 0.25D+0.25S=H 0.25H+T=J 0.25D+0.25S=M 0.25D+0.25H=S 0.25D+0.25H+J+0.25S=T

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