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

2025/12/24 by F. Thomas Bruss, Bruss, F. Thomas, Davy Paindaveine +1 · 1 citation
Mathematics · Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Random Matrices and Applications #Diffusion and Search Dynamics #stochastic dynamics and bifurcation

paper · doi:10.48550/arxiv.2512.21254

Abstract

We study the win rate R_Nd/N_d of a biased simple random walk S_n on ℤ at the first-passage time N_d=inf\n≥ 0:S_n=d\, with p=P[X₁=+1]∈[1/2,1). Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of R_Nd/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 π based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.

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