Gambler's Ruin

The gambler's ruin problem: A gambler starts with \$$a$ and at each step wins \$1 with probability $p$ or loses \$1 with probability $q = 1-p$. The game ends when the gambler reaches \$0 (ruin) or \$$N$ (goal). What is the probability of ruin? Fair game ($p = q =$$\frac{1}{2}$): $$P(\text{ruin}) = 1 - \frac{a}{N}, \qquad P(\text{reach } N) = \frac{a}{N}$$ $$E[\text{duration}] = a(N - a)$$ Biased game ($p \neq q$): $$P(\text{ruin}) = \frac{(q/p)^a - (q/p)^N}{1 - (q/p)^N}$$ Intuition: Even with a slight edge, against an opponent with vastly more resources, you're likely to be ruined. A gambler with \$10 against a casino with \$10{,}000 is doomed even with $p = 0.51$. The asymmetry of resources overwhelms the slight edge. Concrete example (fair game): You have \$20 and want to reach \$100. $P(\text{success}) =$$\frac{20}{100}$ = 20\%. Expected duration: $20 \times 80 = 1{,}600$ steps. Concrete example (biased game): You have \$10, goal is \$20, $p = 0.49$ (house edge of 2%). Let $r = q/p =$$\frac{51}{49}$ \approx 1.0408. $$P(\text{ruin}) = \frac{r^{10} - r^{20}}{1 - r^{20}} = \frac{1.498 - 2.244}{1 - 2.244} = \frac{-0.746}{-1.244} \approx 0.60$$ Even starting halfway to the goal, you're ruined 60% of the time with a tiny house edge! Derivation (martingale method): Wealth is a martingale (fair game) or can be transformed into one (biased game using $(q/p)^{S_n}$). Apply the optional stopping theorem at the stopping time $T = \min(\text{hit } 0, \text{hit } N)$. When to use: Ruin probability analysis in finance and insurance. Position sizing and bankroll management. Understanding why under-capitalized traders fail even with an edge. Classic interview problem in quantitative finance.