Revisiting the name variant of the two-children problem

TL;DR

This paper analyzes a variant of the two-children problem, demonstrating that the probability of the other child being a boy can vary continuously between 0 and 2/3 depending on naming model assumptions, using Schur-concavity.

Contribution

It introduces a natural model for naming in the two-children problem and applies Schur-concavity to analyze how probabilities vary with model parameters.

Findings

Probability can range from 0 to 2/3 based on naming assumptions

Schur-concavity helps understand the dependence on model parameters

The classic 1/2 and 2/3 probabilities are special cases

Abstract

Initially proposed by Martin Gardner in the 1950s, the famous two-children problem is often presented as a paradox in probability theory. A relatively recent variant of this paradox states that, while in a two-children family for which at least one child is a girl, the probability that the other child is a boy is $2/3$, this probability becomes $1/2$ if the first name of the girl is disclosed (provided that two sisters may not be given the same first name). We revisit this variant of the problem and show that, if one adopts a natural model for the way first names are given to girls, then the probability that the other child is a boy may take any value in $(0,2/3)$. By exploiting the concept of Schur-concavity, we study how this probability depends on model parameters.