Answer to 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?

Another little riddle I worked out for myself, so it's probably not that hard...

Before reading the answer can I interest you in a clue?

Right unusually let me tell you the answer first:-

p(t) ≠ f(t)   p(t) = 50%

That is to say the percentage of children that are male is not a function of time and is always 50%. This may seem a little counter intuitive as we know that at some point in time a family could conceivably have 10 girls and one boy, but this is balanced by the fact that half the families will have no girls at all.

The table below is drawn up using the simple rule that of the families who have a new girl one year all will try for a new baby, with half of them having a boy and half having a girl.

End of Year 1Year 2Year 3Year 4Year 5Year Infinity
Total Boys50,00075,00087,50093,75096,875100,000
Total Girls50,00075,00087,50093,75096,875100,000
New Boys50,00025,00012,5006,2503,1250
New Girls50,00025,00012,5006,2503,1250

As I said in the clue, it may help to think of this from the perspective of a midwife who delivers all the babies in the village. Nobody is killing anyone, every child in the village passes through her hands and for her of course there are a 50/50 mix.





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