I was wondering if someone can help me illustrate to an audience what gambler's ruin is by showing it through a transition matrix.
So, for example, showing it for someone trying to get $10,000 starting at $7,000 and betting $1,000 at each betting step
We would have our current fortune, , being and our goal, , being . To make it easy, I can just make the chance of winning, , to be . Then we know
Then what would my transition matrix and vector that I multiply it by be and how would I use matrices to show that the chance of reaching the goal is
or roughly 12% which I figured out using the equation. I want to instead show how to give out that probability using matrices.
Thanks for your help.