A family that would count n members would look like: n= 2parents + number of girls born + 1 son, or more precisely , where k is the number of girls born prior to the birth of a son.
First you set up a random variable X that would count the members of the family. Its values would be , , , , .... You get the pattern, right? where is the number of girls born before the first son.
Then you should assign probabilities to those values.
, where means that the members are girls born prior to a son being born, and he is born with the probability . So you can
So now you have
Next, find the expected value of that random variable:
I think that would be the size of the average family, unless I'm wrong, eh?!