I am trying to calculate the probability that one sequence of coin tosses is more likely to win than the other in a game of Penney's. The sequences are: HTHT and THTT.
So far I've come up with the following:
After some cleaning up we can see that:
$q_2$ and $q_6$ were left out at first, my mistake. Thanks @SteveKass for pointing that out.
But this is where I get stuck. Trying to solve as a system of linear equations gets me no where since I have too many unknown variables.
I want to figure out the possibilities being absorbed in either state $q_4$ or $q_8$ given the current state. Should it really be this difficult?