## Markov Chains

Hi,

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

So..

$P(i,N) = \frac{1-\left(\frac{q}{p}\right)^i}{1-\left(\frac{q}{p}\right)^N}$

We would have our current fortune, $i$, being $7$ and our goal, $N$, being $10$. To make it easy, I can just make the chance of winning, $p$, to be $\frac{1}{3}$. Then we know $q = \frac{2}{3}$

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

$P(7,10) = 0.12414$ or roughly 12% which I figured out using the equation. I want to instead show how to give out that probability using matrices.