Consider the Gambler's Ruin Problem: an infinite Markov process of steps with binary outcomes of plus or minus, each of equal probability. At each step of the process there may be a play of the game.
At each play of the game starting with step 1, the gambler's initial fortune, B, increases by one unit with probability 1/2 or decreases by one unit with probability 1/2. The game ends at step N or sooner (after fewer plays) if the gambler's profit first reaches a Target of +A or a Ruin (total loss) of -B units anytime during the process (which continues beyond N steps).
There are three profit outcomes after step N: -B, +A, or Neither.
Ruin is defined by profit reaching -B for the first time during N plays without first reaching +A. Success is defined by profit reaching +A for the first time during N plays without first reaching -B.
Problem: What is the probability of a. Ruin or b. Success or c. Neither, in terms of A, B, N?
I cannot solve this except by path-counting. Can a.,b.,c. be solved in a closed form by Markov chains?
This problem is also called a random walk of a particle in finite time with two absorbing barriers.