Gender Ratio in a Society with Stopping Rule

This question describes a population where families follow a simple stopping rule in childbearing, and asks about the resulting gender proportions across the whole society. The setup highlights the contrast between intuition at the family level (where behavior is clearly skewed toward achieving a particular outcome) and what happens when you aggregate over many independent families and many births. It is a classic conditional probability and reasoning puzzle, often seen in interviews for roles that require probabilistic thinking, such as trading and risk roles at banks and quantitative hedge funds.

The problem leans heavily on understanding independent Bernoulli trials, geometric-type stopping rules, and long-run frequency interpretations of probability. It indirectly probes whether the candidate confuses a mechanical rule for a genuine change in probabilities. Interviewers look for clean reasoning about sample spaces and expectations, and for an explicit separation between the distribution of family sizes and the overall distribution of outcomes. They are also checking comfort with the law of large numbers and the idea that optional stopping of an i.i.d. process does not, by itself, create bias.

What it tests

Whenever you have a process where each trial is independent and identically distributed, and a 'stopping rule' is based on the outcome of those trials, the overall long-term proportions of outcomes are determined by the underlying probabilities, not the stopping rule. This is a consequence of the law of large numbers and the memoryless property of independent trials. The stopping rule can affect the distribution of the number of trials per sequence (such as family size), but it cannot bias the proportion of outcomes in the aggregate. The key is that each event (here, each birth) is unaffected by prior events or by the decision to stop, so the expected frequencies reflect the base probabilities. This principle holds for any process with independent, identically distributed outcomes and a stopping condition based on a particular result.