Clue for Riddle #4: Mythical City Population of Boys

4. A mythical city contains 100,000 married couples but no children. Each family wishes to "continue the male line", but they do not wish to over-populate. So, each family has one baby per annum until the arrival of the first boy. For example, if (at some future date) a family has five children, then it must be either that they are all girls, and another child is planned, or that there are four girls and one boy, and no more children are planned. Assume that children are equally likely to be born male or female.

Let p(t) be the percentage of children that are male at the end of year t. How is this percentage expected to evolve through time?

The first thing to realise is that if you toss a coin 9 times and get 9 heads the tenth toss is still equally likely to be heads or tails.

For the second point, without giving too much away. Don't look from the families' perspective, but imagine the village has only one midwife. This midwife would see all the births in the village. In a given period what would she expect the mix of boys to girls to be?

A bit counter intuitive this one.


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