Mathematics of Family Planning in Talmud

TL;DR

This paper explores the mathematical principles behind Talmudic family planning rules, revealing that despite different religious guidelines, gender ratios remain constant and equal to birth odds, indicating no gender discrimination.

Contribution

It introduces a mathematical analysis of Talmudic family planning rules, demonstrating that gender ratios are unaffected by religious constraints and equal the natural birth odds.

Findings

Gender ratios are constant across different rules.

Ratios are equal to natural birth odds.

Religious rules do not discriminate by gender.

Abstract

Motivated by the commitments from the Talmud in Judaism, we consider the family planning rules which require a couple to get children till certain numbers of boys and girls are reached. For example, the rabbinical school of Beit Hillel says that one boy and one girl are necessary, whereas Beit Shammai urges for two boys. Surprisingly enough, although the corresponding average family sizes differ in both cases, the gender ratios remain constant. We show more that for any family planning rule the gender ratio is equal to the birth odds. The proof of this result is given by using different mathematical techniques, such as induction principle, Doob's optional-stopping theorem, and brute-force. We conclude that, despite possible asymmetries in the religiously motivated family planning rules, they discriminate neither boys nor girls.