Male Child Ratio Under One-Son Rule

This question considers a population-wide birth policy where families keep having children until a condition involving a male child is met, then stop. You are asked to track how the proportion of boys among all children evolves over calendar time, not just per family. The setup forces you to think about overlapping birth histories for many families, some that stop early and others that continue much longer, and to translate a family-level rule into aggregate statistics as the years go by. It is a classic style of probability puzzle that could come up in interviews for junior quant or data science roles to test intuition about stochastic processes and large populations.

Solving it leans on modeling births as independent Bernoulli trials with a fixed success probability and recognizing that the stopping rule is history-dependent but outcome-neutral at each trial. The interviewer wants to see whether you distinguish between the distribution of family sizes and the global gender ratio, and whether you can argue about convergence over time using law-of-large-numbers style reasoning rather than ad hoc simulation. Clear conditioning, careful counting across cohorts, and the ability to articulate why the stopping rule does or does not bias long-run proportions are central.

What it tests

When a process is governed by independent random events with fixed probabilities, and a stopping rule is based solely on the outcome of those events, the long-run proportions of outcomes are determined by the underlying probabilities, not the stopping rule. This is because each trial is memoryless: the probability of a boy or girl at each birth remains $1/2$, regardless of previous outcomes or when the process stops. The stopping rule may affect the distribution of family sizes, but not the overall ratio of boys to girls in the population. This principle is a consequence of the law of large numbers and the independence of each trial, ensuring that aggregate proportions converge to the underlying probabilities as the number of trials grows. The key is that the stopping condition does not introduce any bias into the outcome of each individual event.