We present below a paradox popularized by Martin Gardner in October 1959 in his “Mathematical Games” column in Scientific American, under the title “The Two Children Problem.”

We will analyze a probabilistic problem related to a hypothetical policy where each family in a country has children until they have a boy, at which point they are required to stop. Our goal is to determine the resulting proportion of boys and girls in this country.

The Hypothetical Policy: Link to heading

  • Each family continues to have children until a boy is born.
  • For each child, the probability of being a boy or a girl is 50%, independent of the others, including twins.

Structurally, the proportion of families with at least one boy will tend towards 100%, as opposed to 50% before the policy is adopted.

Intuitively, one might think that such a policy would favor more boys than girls. However, the actual proportion of boys to girls remains constant.

Theoretical Explanation Link to heading

Each family will, on average, have one boy (since they stop once a boy is born). The number of girls per family follows a geometric distribution.

The probability distribution for the number of girls in a family that stops at the first boy is as follows:

  • The probability of having 0 girls (a boy on the first try) is 50%.
  • The probability of having 1 girl followed by a boy is 25%.
  • The probability of having 2 girls followed by a boy is 12.5%, and so on.

According to the law of large numbers, as the number of families increases, the average number of boys per family converges almost surely to 1, and the average number of girls per family converges in probability to 1.

Thus, at the population level, the boy-to-girl ratio remains 1:1.

Probability theory can only estimate that, on average, each family is likely to have one girl, but we are certain that no family can have more than one boy. We may encounter families with ten girls, but never families with two or more boys.

To apply the Law of Large Numbers to this reproductive model, a Minimum Viable Population (MVP) is necessary to ensure the viability of the process.

Reproductive Process Viability Link to heading

In the specific context where the reproductive process continues only if the population has at least 8 men and 16 women, and assuming the population starts with exactly this minimal configuration, we can assess the long-term viability of the process under the hypothetical policy.

This involves evaluating the probability that, due to random fluctuations inherent in the reproductive process, the population will not fall below these critical thresholds in future generations.

Context Link to heading

  • Each person continues to reproduce until a boy is born (with probability p = 0.5), or until a cap is reached (maximum 40 children per woman, though in practice a more reasonable limit applies).
  • If the total number of males falls below 8 or the number of females below 16, the process stops (extinction).
  • Every woman reproduces, given a pair.

We have slightly altered the policy in the sense that families remain intact and do not separate for subsequent reproduction.

We are guaranteed 16 reproductions if the initial 8 men each produce only boys and thus stop. An additional 8 reproductions will follow.

There is a probability of $2^{-16}$ that the process ends at this stage.

The diagram below illustrates a Markov chain used to model the transition probabilities between states (in percentages), where each state represents the number of girls remaining in the reproductive process. State 0 is absorbing and indicates that there are no more girls — meaning the process has ended.

This approach comes with two complementary interpretations:

  • In dynamics, it shows how the number of girls decreases over time (after each generation), depending on the binomial distribution of new children (with a 50% chance of being a girl);
  • In retrospect, the diagram allows us to estimate the probability that, starting from a given number of girls, the process will lead to extinction (state 0) within a certain number of steps.

The probability that the process ends (extinction) is quite low: $$\sum_{k=0}^{16} \binom{16}{k} \cdot 2^{-16} \cdot 2^{-k} \approx 1%$$

The scenario in which all 8 pairs fail to produce a boy in 320 reproduction attempts is extremely unlikely, although theoretically possible. This corresponds to a residual risk of reproductive inviability.

There is a slightly greater than 1% chance that the Law of Large Numbers cannot be applied in this context. However, this hypothesis cannot be rejected at a 98% confidence level.

This paradox is characteristic of the entire field of Stochastic or Actuarial Mathematics! test No need to understand all of this—just find the best certified actuary who will handle everything for you!