Suppose they said, “I used to be part of this club that I liked, but they had a rule that if anyone ever missed the weekly Tuesday meeting, they were kicked out, with no exceptions. I really liked that club, but being kicked out was worth it, worth it a thousand times over, to be there as my wife gave birth to my son (which happened on a Tuesday).” and at some other point imply that the two of them have two kids?
(This contrived example is because the first example I thought of implied that their other child was probably not born on a Tuesday. This example avoids this because if the other child came second, the same reason for it being relevant if it was on a Tuesday doesn’t apply. Uh… hm, actually, not sure if this fixes the issue…)
P([they have two sons] | [they have two kids] & [they have a son born on a Tuesday]) = P([they have two kids who are both boys] & [at least one of their two kids was a boy and born on a Tuesday] | [they have two kids])/P([at least one of their two kids was a son born on a Tuesday] | [they have two kids])
If two sons, the chance that at least one was born on a Tuesday is 13/49 . If they have one son, the chance that have a son born on a Tuesday is 7/49.
So, given have two kids, chance of both boys and at least one born on a Tuesday is (1/4) * (13/49) = 13/196.
Given two kids, chance that at least one son born on a Tuesday is (2/4)(7/49) + (1/4)(13/49) = 27/196
P([they have two sons] | [they have two kids] & [they have a son born on a Tuesday]) = P([they have two kids who are both boys] & [at least one of their two kids was a boy and born on a Tuesday] | [they have two kids])/P([at least one of their two kids was a son born on a Tuesday] | [they have two kids])
If two sons, the chance that at least one was born on a Tuesday is 13/49 . If they have one son, the chance that have a son born on a Tuesday is 7/49. So, given have two kids, chance of both boys and at least one born on a Tuesday is (1/4) * (13/49) = 13/196.
Given two kids, chance that at least one son born on a Tuesday is (2/4)(7/49) + (1/4)(13/49) = 27/196
So, (13/196)/(27/196) = 13/27