Here are two puzzles, superficially similar, but different answers.

A. Alf meets Bert, and asks him how many children he has, and of what sex.
The puzzle: if Bert has two children and at least one of them is a boy, what are the odds that he actually has two boys?

B. Alf now asks the same questions of several more people, until he gets to Charlie.
The puzzle: if Charlie has two children and at least one of them is a boy, what are the odds that he actually has two boys?

C. Alf goes back to Bert, and asks him what days any boys were born on.
The puzzle: if Bert has two children and at least one of them is a boy born on a Tuesday, what are the odds that he actually has two boys?

D. Alf now asks the same questions of several more people, until he gets to Dave.
The puzzle: if Dave has two children and at least one of them is a boy born on a Tuesday, what are the odds that he actually has two boys?

This strange puzzle and/or variants of it was originally set by Martin Gardner. The surprising thing is that the exact same question can have two different answers depending on the assumed context from which the odds should be calculated. There are enough clues here to make the answers fairly obvious.

This puzzle is unsatisfactory, and in some ways it was intended to be so. You don’t have to agree with my answers, but if you don’t at least know why.

Odds may be defined as the number of favourable outcomes as a fraction of the total number of outcomes, all outcomes being equally likely. The intention was as follows.

A. Total outcomes: 2 (other child could be boy or girl). Favourable: 1. Answer 1/2.
B. Total outcomes: 3 (boy/boy, boy/girl or girl/boy). Favourable: 1. Answer 1/3.
C. Same as A. Answer: 1/2.
D. Total outcomes: 27 (boy Tue/boy any=7, boy not Tue/boy Tue=6, boy Tue/girl any=7, girl any/boy Tue=7). Favourable: 13. Answer 13/27.

The problem is not the maths, but the ambiguous language used to frame the questions. In particular, it is very difficult to be sure how many total outcomes were possible at the time the question was asked. I’m happy for anyone to disagree with my interpretation, because the point was really to show the gap between the verbal question and the underlying maths.