Win rates at first-passage times for biased simple random walks

We study the win rate $R_{N_d}\mathrm{/}{N}_d$ of a biased simple random walk $S_n$ on $\mathbb{Z}$ at the first-passage time $N_d=\inf \{n\ge 0:S_n=d\}$, with $p=P[X_1=+1]\in [1\mathrm{/}2,1)$. Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of $R_{N_d}\mathrm{/}{N}_d$ along with monotonicity properties in the threshold $d$ and the bias $p$. We also provide closed-form expressions and use them to design unbiased coin-flipping estimators of $\pi$ based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.