when we hear the “I have two children, at least one of whom is a boy” part, we set the probability of two boys to 1⁄3 because the possibilities {(boy, girl), (girl, boy), (boy, boy)} are a-priori equally likely
Why is this the most common assumption? This never made much sense to me whenever I’ve encountered this problem.
It’s much more intuitive to think about the scenario as:
2xB
1xB 1xG
2xG
Rather than:
BB
BG
GB
GG
And to come to an answer of 1⁄2 instead of 1⁄3. The question doesn’t state anything about the children’s gender being related to the order they were born.
Good point.
You’ve made me realize that I’ve misrepresented how my intuitive mind processes this. After thinking about it a bit, a better way to write it would be:
Child 1: P(B) = 1⁄2, P(G) = 1⁄2
Child 2: P(B) = 1⁄2, P(G) = 1⁄2
Combined as unordered set {Child 1, Child 2}
The core distinction seems to be to be if you considered it an unordered set or an ordered one. I’m unsure of any way to represent that in easy to read text format, the form written above is best I’ve got.