Sally's Bets

Despite first appearances, none of Sally's three wagers have the same odds.

Sally's first wager is closest thing to a fair bet she offers. The fact that the older child is a girl has no bearing on the sex of the younger child. Your odds of collecting on this bet are basically fifty-fifty.¹

Sally's second wager is less than fair. While the youngest child of any random family is equally likely to be a boy as a girl, Sally has carefully chosen a subset of the population to help her out. There are four possible combinations of two children families: both boys, both girls, older boy/younger girl, and older girl/younger boy. Her question eliminated the both boys combination. Of the remaining three possibilities, two will give her a win. Unless the couple she has asked has two girls, the bet is in her favor.

If you are unconvinced, consider this: What if the couple had responded "No, neither of our children is a girl?" Sally would have been unable to offer the wager she did (at least reasonably), and would have had to move on to the next couple.

If you took Sally's third wager, you deserved what you got. What are the odds that Sally doesn't know the sex of her own brother?

Footnote 1: In the United States, females comprise 51% of the population, which might give you a small edge in this wager. However in the younger age brackets, males make up over half the population. So your exact odds would be slightly influenced by the ages of the couples present, which was not stated in the problem. Statistical fluxuations due to age, race, and nationality of those present, however, are unlikely to offset the odds much in your favor.