Moe and Joe flip a slightly unbalanced coin until one or the other loses all his money. On a single flip Joe wins with probability .49. Joe starts with $8 and Moe starts with $4.
1. Find the probability Joe will win all of Moe's money.
2. Write out the transition matrix P for this game (where the state of the system is the amount of Joe's money).
3. Find the two step transition matrix.
4. Does P^n have a limit as n increases without limit and, if so, what is it?