Revisiting the Name Variant of the Two-Children Problem

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.

KEYWORDS:

• Majorization
• Paradoxes in probability theory
• Schur-concavity/convexity
• Stochastic modeling

Acknowledgments

The authors would like to thank the Editor, the Associate Editor and two referees for the careful reviews of the manuscript and insightful comments and suggestions. The present work results from exchanges following a talk of the Altaïr conference cycle in Brussels; we would like to thank the organizers.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 In the rest of the article, “name” will throughout stand for “first name”.

Additional information

Funding

This research is supported by the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles and by the Fonds Thelam from the Fondation Roi Baudouin.