Random Walk Boundary

Solution

Answer: 27.3%

After \(8\) steps there are \[ 2^8 = 256 \] possible step sequences. We want the probability that the walk never goes below \(0\) at any time. Because the walk begins at \(0\), the first step must be \(+1\); otherwise the walk immediately goes to \(-1\). After this first step we are at position \(1\) with \(7\) steps remaining. A key fact from the reflection principle / Catalan counting is that the number of walks of length \(2n\) that start at \(0\) and never go below \(0\) is \[ \binom{2n}{n}. \] However, when the walk ends above \(0\), only a portion of those paths are valid. The correct count of non-negative walks of length \(2n\) is \[ \binom{2n}{n}. \] For \(n=4\), \[ \binom{8}{4} = 70. \] So there are \(70\) walks of length \(8\) that never go below \(0\). Since all \(256\) walks are equally likely, the probability is \[ \frac{70}{256} = \frac{35}{128} \approx 0.2734. \] So the probability that the walk never dips below \(0\) during the \(8\) steps is about \(27.3\%\).