On stopping rules and the sex ratio at birth

One popular stopping rule intended to increase the rate of male births is to keep having babies until the first male is born. We show that such a stopping rule does not change the sex ratio at birth when the probability p that a birth produces a male is constant across a population. We show, however, that when p varies across couples, as evidence suggests that it does in the human population, with mean around 0.515 and standard deviation 0.05, then such a stopping rule tends to favour female births and the correct rule should be to stop procreating until the first female is born. But we also show that it does not seem that such stopping rules, even if favouring male births, can account for the dramatic increase in the rate of male births registered in some countries. Most of these conclusions have appeared in some form or other in various studies, but they are scattered across the literature and are very often presented using some heavy mathematical techniques. In this paper, we try to bring together these ideas and make them more accessible by analysing these stopping rules using the simplest mathematical tools possible.