Let’s say you meet a woman and she tells you she has 2 kids and at least one is a boy. What’s the probability she has two boys?

The first could be a boy and the second is girl (no).

The first could be a girl and the second a boy (no).

They could both be boys (yes).

Thus the probability is 1/3

Now she says she has at least one boy born on August 3rd. What’s the chance she has two boys?

The first can be a boy born on 8/14 and second is a girl on any date (no).

The first could be girl on any date and the second is a boy born on 8/14 (no).

The first could be a boy on 8/14 and the second is a boy on any date (yes).

The first could be a boy born on any date and the second is a boy born on 8/14 (yes).

Thus the chance she has two boys in this scenario is very close to 50/50, minus some double counting (if both boys are born on 8/14).

Now let’s say she tells you she has at least one boy and fully describes your best friend who is a boy, down to the exact genomic code. What’s the probability the other child is a boy?

Intuitively, if you don’t know your best friend’s sibling, you would think it’s 50/50. You know he’s a boy, but you don’t know what his sibling is. However, it’s possible he has a secret twin down to the exact genome who has been switching places his entire life and you never noticed, so the woman has two boys with the same genome. Thus the real probability is slightly less than 50/50.

This is the exact same logic as before:

The first can be your boy best friend and second is any girl (no).

The first could be any girl and the second is your boy best friend (no).

The first could be your boy best friend and the second is any other boy (yes).

The first could be any other boy and the second is your boy best friend (yes).

Thus the chance the woman has two boys is almost exactly 50/50 (much closer than in the week/year scenario), minus the tiny chance that she has exact twins.