Gambler's Coin

Solution

Answer: 0.25

You start with \(50\) dollars, the absorbing lower boundary is \(0\) (ruin), and the absorbing upper boundary is \(200\). Let \[ p_i = P(\text{reach }200 \text{ before }0 \mid \text{start with } i). \] For a fair game, the process satisfies \[ p_i = \frac{1}{2}p_{i+1} + \frac{1}{2}p_{i-1}. \] Rearranging gives \[ p_{i+1} - p_i = p_i - p_{i-1}. \] This shows the differences are constant, so \(p_i\) is a linear function of \(i\). Write \[ p_i = ai + b. \] Use the boundary conditions. If you start with \(0\) dollars, you are already ruined: \[ p_0 = 0. \] Thus \[ b = 0. \] If you start with \(200\) dollars, you have already reached the target: \[ p_{200} = 1. \] Substituting, \[ 1 = 200a. \] \[ a = \frac{1}{200}. \] Therefore \[ p_i = \frac{i}{200}. \] Starting with \(i = 50\), \[ p_{50} = \frac{50}{200}. \] \[ p_{50} = 0.25. \]