Matthen2 posted this problem on twitter and I thought it was interesting. If families all have children until they have exactly one daughter, on average how many children of either sex will they have? In this post I play with the probabilities involved.
In order to answer this question let us consider what is the probability that a family has children. Since they have children until they have a daughter, and each time they have a child the probability of it being female is , the probability of having children is . In order to determine the average number of children per family we must therefore evaluate the following sum:
The first terms of the sum are: . The answer to this sum is readily found by asking WolframAlpha (the answer is 2). Is it possible to determine the summation with good old pen and paper? The solution is not immediately obvious as the summation is neither a geometric, nor an arithmetic sum. In order to make the sum, let us make a table of the table above:
It is clear that this sum is the same as:
OOOOH! Now the columns are geometric series and they can be easily summed. The resulting values are:
So the average number of babies per family is two, since each family has exactly one daughter, this means that on average each family has one son. So to achieve a “perfect” birth rate of 2, all we have to do is have enough babies to have exactly one daughter. But never, ever have two daughters.
PS: The previous argument requires families to be prepared to have infinite number of babies, if necessary. What if each family decided to have at most babies? The the average number of girls in each family would b:
which is the probability of having boys. The number of boys would be:
Where the first second represents the average of the number of boys in families with one girl, the first term is the probability of having boys weighted by the number of boys. With Wolfram‘s help:
So if people are going to follow the above rules, but only want to have – say – 4 children, the number of children per family will be $latex~2~(1-1/16)=15/8$. And still equality of boys and girls!