 Gambler's ruin

Gambler's ruin is the absorption problem for a fortune that changes by one unit per game. Let $S_n\in \{0,1,\dots ,N\}$, with independent wins of Probability $p\in (0,1)$ and losses of Probability $q=1-p$. Stop when the fortune reaches zero or the target $N$. Zero is ruin; reaching $N$ is success.

The success Probability $h_i={\mathbb{P}}_i(T_N<T_0)$ has boundary values $h_0=0$, $h_N=1$ and satisfies $h_i=p{h}_{i+1}+q{h}_{i-1}$. Solving this second-order recurrence givesFinite-interval exit times have geometric tails, so absorption occurs almost surely and ruin has Probability $1-h_i$. The fair case is linear because the fortune itself is harmonic; the biased case uses the harmonic exponential ${\rho }^i$.

For a fair game with initial fortune three and target ten, success has Probability $3/10$. The Expected duration of symmetric gambler's ruin is $i(N-i)$, giving 21 steps in this example. The same local first-step method produces expected durations, discounted absorption probabilities and conditional durations; their different boundary conditions encode different questions about the same paths.

Gambler's ruin is a concept from probability theory and statistics that models a gambling scenario where a gambler continues to gamble until either they lose all their money or reach a predetermined target amount. It is often used to illustrate the principles of random walks and the behavior of stochastic processes. In a typical setup, a gambler starts with a certain amount of capital and bets on a game with a fixed probability of winning or losing.