Six people (Abbey, Blake, Chulin, Darian, Emily, and Tony) play catch. If Abbey

Question

Six people (Abbey, Blake, Chulin, Darian, Emily, and Tony) play catch. If Abbey has the ball, she is equally likely to throw it to Blake, Darian, Emily, and Tony. If Blake has the ball, he is equally likely to throw it to Abbey, Chulin, Emily, and Tony. If Emily has the ball, she is equally likely to throw it to Abbey, Blake, Darian, and Tony. If either Chulin or Darian gets the ball, they keep throwing it back and forth to each other. If Tony gets the ball, since he is terrible at throwing, he ends up throwing the ball into the neighbor’s yard, where it is promptly eaten by their dog. (a) Find the transition probability and classify the states of the chain. (Hint: don’t give the dog it’s own state.) (b) Suppose Abbey has the ball at the beginning of the game. What is the probability Tony’s neighbor’s awful dog eats the ball?

Explanation / Answer

Solution

Let A,B,C,D,E and T be the state of throwing a ball

(b) A has ball at the beginning so,

(1 0 0 0 0 0)*(matrix)

=(0 0 0 0 0 0.25)

probability = 0.25

A B C D E T A 0 0.25 0 0.25 0.25 0.25 B 0.25 0 0.25 0 0.25 0.25 C 0 0 0 1 0 0 D 0 0 1 0 0 0 E 0.25 0.25 0 0.25 0 0.25 T 0 0 0 0 0 1

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