Risk of Ruin for normally-distributed games Classic ruin theory assumes that the income is constant and that only the losses are random with an underlying distribution. Suppose we want to determine the risk of ruin for a game (for example poker), where also the winnings are random. Let $\psi(u)$ denote the risk of ruin, starting from initial surplus $u$. We assume that the winnings/losses in each game follow a normal distribution with mean $\mu>0$. The game is played until the player is broke or his surplus goes to infinity. Classical ruin theory doesn't seem to apply because of the non-constant income. I am grateful for any advice on how to approach this problem.

According to risk of ruin wikipedia page,

The formula for risk of ruin for such setting can be approximated by

$$\left( \frac{2}{1+\frac{\mu}{r}}-1 \right)^{u/r}=\left(\frac{1-\frac{\mu}{r}}{1+\frac{\mu}{r}} \right)^{u/r}$$

where $$r=\sqrt{\mu^2+\sigma^2}$$

It is described that the approximation formula is obtained by using binomial distribution and law of large numbers.

I have written the formula in the form of proposed by Perry Kaufman

$$\left( \frac{1-\text{edge}}{1+\text{edge}}\right)^{\text{capital units}}$$