For those that aren't familiar with Gary Foshee's probability puzzle/paradox from 4-5 years ago, you can find an analysis here:
While this puzzle has been discussed here on Stack Exchange:
The Tuesday Birthday Problem – why does the probability change when the father specifies the birthday of a son?
(two or three times in fact), the focus has been on the solution and not the LOGIC/MECHANICS behind why the solutions between the two statements below are different.
Statement 1: "I have two children. One of them is a boy. What is the probability I have two boys?" — OK, the explanation makes sense, it's 1/3.
Statement 2: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?" — Again, the explanation makes sense, it's 13/27
What are the mechanics behind the change in probability? Is it because the more specific he is about one of his children the less chance it could be either child he is referring to as opposed to one in particular? So there's some kind of probability "overlap" that is removed (sorry, I don't know a better way to put it).
I've seen this explained similarly here (though still not quite as satisfactorily as I would like):
By adding specific information "You've effectively told people about one individual, not told them that one of your children is a member of a category."