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Thank you all for playing along! I’ve really enjoyed watching the discussion play out. Since this market has attracted more traders and mana than any other I’ve done, I’m guessing the feeling is mutual. I’ll have to come up with more like this!

Special thanks to Pat Scott for pointing out the hilarious typo and to Fion for pointing out that my imprecise wording didn’t actually change the answer of the yes/no question.

I found this puzzle years ago in a professional engineering magazine. I was indignant that an engineering publication would get such a basic probability concept wrong. I mean, obviously the sex of one child is independent of the other! I sat down and did the bayesian calculation in preparation for my indignant letter to the editor only to discover that I was wrong!

Fion did a great job of explaining the logic- that the other child is a boy in two out of the three equally likely scenarios.

The Bayesian calculation was mentioned by Wamba Ivanhoe. The form you usually see, P(A|B) = P(B|A)P(A)/P(B), isn’t all that helpful since P(B|A) is just the mirror of the question you are puzzling over. You’ll want the form P(A|B) = P(B and A)/P(B). With two children, the probability of a girl and a boy is 1/2. The probability of a girl is 3/4*. 1/2 divided by 3/4 is 2/3.

This is actually a classic enough puzzle to have its own wikipedia page. https://en.m.wikipedia.org/wiki/Boy_or_Girl_paradox

I’d recommend the section on variants. The subsection there on additional information about the child I found to be particularly interesting.

*probability of at least one girl = 1- probability of no girls in two= 1 - P(boy)^2 = 1-(1/2)^2 = 1 - 1/4

@VickiWilliams More puzzles with propability, plz.

@VickiWilliams Thanks for making the market. This question really messed with me first time I heard it! I hope I didn't spoil anybody's fun by posting my answer. I'm not sure whether it came from a place of wanting to be helpful and honest, or if I was paranoid there would be a trick somehow and I'd get the rug pulled any minute. (Hence my limit orders at 1% and 2%...)

Good fun, though, and I'm pleased to be made aware of the Wikipedia page!

The resolution will assume a 1:1 sex ration for the human population.

calling out the typo here not because I want to be pedantic but because it's hilarious

P(Boy and Girl) / P(at least one girl)

@ShitakiIntaki oh nooo this is the mathematical proof that I'm wrong. oops, I should have read more! fascinating and unintuitive

@ShitakiIntaki I think another good intuitive way to think about it is similar to “your friends probably have more friends than you”—that is, given that someone is your friend, they’re more likely to have many friends. Or the extreme case where there’s one ultra popular people and 50 people only friends with that person. Or in this sense given that someone has two children and one of them is a girl that means it’s slightly more likely that the second one is also.

Well, I'd say that lesson is worth the twenty mana I spent to learn it.

3 possibilities:

1. Older child is a girl; younger child is a girl

2. Older child is a boy; younger child is a girl

3. Older child is a girl; younger child is a boy.

Of the three possibilities, 2 of them have the other child being a boy, so the probability that the other child is a boy is 2/3 =/= 0.5.

@Fion there's an alternative, very pedantic interpretation of the question where we note that you said "one" child is a girl, and we assume this means "exactly one", in which case the probability that the other child is a boy is 1. Obviously this is a silly interpretation, but either way the answer to the question you asked is "no".

@Fion The odds of the second child being born male don't depend on the sex of the firstborn.

@Schwabilismus true, but the question didn't say "conditional on the first child being female", it said "conditional on one child being female".

If the question specified birth order then the answer is yes. If it doesn't, the answer is no.

@Fion another child being a specific sex isn't a condition here. It doesn't influence the propability of the second child's sex.

@Schwabilismus we're not talking about the second child, we're talking about the "other" child. It could be the second or the first

@Fion This market is interesting. Either i am totally wrong and about to learn a "valuable lesson", or i am about to make a boatload of money. Either way it is bedtime where i live. I'll reply tomorrow.

@Fion the way you've explained this it feels like a monty hall problem. but I'm not convinced it is! hmmm debating whether I should bet or do math.

@Stralor smol bet it is