dicing-with-death-chance-risk-and-health (Stephen Senn)

Mr Brown has exactly two children. At least one of them is a boy. What is the probability
that the other is a girl?

What could be simpler than that? After all, the other child either is or is not a girl.
I regularly use this example on the statistics courses I give to life scientistsworking
in the pharmaceutical industry. They all agree that the probability is one-half.

So they are all wrong. I haven’t said that the older child is a boy. The child I
mentioned, the boy, could be the older or the younger child. This means that Mr
Brown can have one of three possible combinations of two children: both boys, elder
boy and younger girl, elder girl and younger boy, the fourth combination of two girls
being excluded by what I have stated. But of the three combinations, in two cases the
other child is a girl so that the requisite probability is 2/3 …

This example is typical of many simple paradoxes in probability: the answer is
easy to explain but nobody believes the explanation. However, the solution I have
given is correct. Or is it? That was spoken like a probabilist. A probabilist is a
sort of mathematician. He or she deals with artificial examples and logical connections
but feel no obligation to say anything about the real world. My demonstration, however,
relied on the assumption that the three combinations boy–boy, boy–girl and girl–boy are
equally likely and this may not be true. The difference between a statistician and a
probabilist is that the latter will define the problem so that this is true, whereas
the former will consider whether it is true and obtain data to test its truth.