Transition Matrix - homework check

52. **Three people -- John, Joan, and Kim -- throw a ball to each other. There is a probability of 1/3 that John will throw the ball to Joan. There is a probability of 1/2 that Joan will throw the ball to Kim. There is a probability of 1/4 that Kim will throw the ball to John. **

b) Assuming that the ball starts with Joan, what is the probability that she will have it back after 2 throws?

_____ ___ ___

[__0 _____ 1/3 ____ 2/3 ]2 (this matrix squared)...

[ _ 1/2 ____ 0 ______ 1/2 ]

[__ 1/4 ____ 3/4 _____ 0 ]

..which gives you

[__1/3_____ 1/6 ____ 1/6 ]

[ _ 1/8 ____ 13/24___ 1/3 ]

[__ 3/8 ____ 1/12_____13/24 ]

Therefore, there is a 13/24 probability that Joan will have the ball.

Second part of question.. Assuming that the ball starts with Kim, what is the probability that Joan will have it after 3 throws??

THANKS