A gambler starts with `i` coins. Each round independently increases the fortune by one with probability `p` and decreases it by one with probability `1-p`. Play stops at `0` or `N` coins. Derive the probability of reaching `N` before `0` for `0 <= i <= N`. Give both the fair case `p = 1/2` and the biased case `p != 1/2`, including boundary conditions. ### Constraints & Assumptions - `0 < p < 1` and `N` is a positive integer. - Rounds are independent and the step size is exactly one coin. - The stopping boundaries are absorbing. ### Clarifying Questions to Ask - Is the requested quantity a hitting probability or expected duration? - Are the win probabilities constant over time? - What should the formula return at `i = 0` and `i = N`? ```hint Write a recurrence For an interior fortune, condition on the next round and solve the resulting second-order difference equation. ``` ### What a Strong Answer Covers - The recurrence and two boundary conditions. - The linear fair-case solution and geometric biased-case solution. - Correct limiting or sanity checks as `p` approaches one half and at both boundaries. ### Follow-up Questions - How would you derive the expected absorption time? - What changes if wins add two coins but losses remove one? - How does the probability behave as `N` grows when `p < 1/2`?