Weighted Birth Stopping Rules with Varying Thresholds

2026/08/03 by Rui Viana
Mathematics · #math.PR

paper · pdf

11 pages, 3 figures

arxiv created 2026/08/03 · arxiv updated 2026/08/05

Abstract

We study a class of stopping-rule questions in which each of \(n\) families continues having children until its number of girls first equals \(d\) times its number of boys plus a positive integer threshold \(k_i\). After all families have stopped, we ask for the expected proportion of boys and the expected boy-to-girl ratio in the combined population. We obtain exact series for both expectations, allowing the probabilities of a boy and a girl to be unequal. The expected-proportion formula recovers known results for the fair first-girl rule and for the unweighted girl-majority rule. To our knowledge, these summations for the general weighted rule with varying thresholds are new. We give two elementary proofs, one bijective and one probabilistic.

Citations

• Win rates at first-passage times for biased simple random walks
• Unparalleled instances of prolifickness, random walks, and square root boundaries