- #1
crazy_craig
- 9
- 0
Homework Statement
Six children (Dick, Helen, Joni, Mark, Sam, and Tony) play
catch. If Dick has the ball he is equally likely to throw it to Helen,
Mark, Sam, and Tony. If Helen has the ball she is equally likely to
throw it to Dick, Joni, Sam, and Tony. If Sam has the ball he is
equally likely to throw it to Dick, Helen, Mark, and Tony. If either
Joni or Tony gets the ball, they keep throwing it to each other. If
Mark gets the ball he runs away with it. (a) Find the transition
probability and classify the states of the chain. (b) Suppose Dick
has the ball at the beginning of the game. What is the probability
Mark will end up with it?
I only need help with (b). I created the transition matrix as below.
Homework Equations
So, by "ends", I'm assuming we want the lim n→∞ pn(x,y) = π(y), but I don't
think that this chain converges (by raising this transition matrix to high powers), so there must be another way to solve this. It seems easy and I'm probably overlooking something. I've spent waay too long on this, so I'm asking for some help and a nudge in the right direction.
Thank you very much!
The Attempt at a Solution
\begin{array}{cc}&D & H & J & M & S & T \\
D& 0&.25&0&.25&.25&.25 \\
H&.25&0&.25&0&.25&.25\\
J&0&0&0&0&0&1\\
M&0&0&0&1&0&0\\
S&.25&.25&0&.25&0&.25\\
T&0&0&1&0&0&0\\
\end{array}