Gambler's Ruin with Asymmetric Probabilities and Hitting Time

Scenario

A gambler starts with fortune \k$and repeatedly wagers$$1$on successive independent rounds, winning$$1$with probability$p$and losing$$1$with probability$q = 1 - p$. The gambling process halts when the player either reaches$$0$(ruin) or attains target wealth$$N$($N > k > 0$).

Your Task

1. Formulate the second-order linear recurrence relation for the probability of reaching target $N$ ($P_k$).
2. Derive the closed-form analytical formula for $P_k$ for both fair ($p = 0.5$) and biased ($p \ne 0.5$) regimes.
3. Derive the expected duration (hitting time) $E_k$ of the game.
4. Compute $P_k$ and $E_k$ for $k = 20, N = 50, p = 0.48$.

Constraints

• Boundary conditions: $P_0 = 0, P_N = 1$ and $E_0 = 0, E_N = 0$.

Examples For $p = 0.5$, $P_k = k / N$ and $E_k = k(N - k)$.