In the Gambler's Ruin problem, the probability $P(k)$ of reaching $n$ dollars (winning) before going broke (reaching $0), starting from $k$ dollars, satisfies: $P(k) = \frac{1}{2}P(k-1) + \frac{1}{2}P(k+1)$ with boundary conditions $P(0) = 0$ and $P(n) = 1$. This rearranges to the linear recurrence $P(k+1) - 2P(k) + P(k-1) = 0$. Which is the closed form for $P(k)$?