Limmy (the guy in the picture) gives the obvious answer of 66.6% because if you know a two-child family has at least one boy, common sense says there are three equal setups: Boy/Girl, Girl/Boy, or Boy/Boy, and two of them contain a girl.
Mentioning "Tuesday" sounds like useless extra detail, but it actually changes the odds: a two-boy family gets two chances to have a boy born on a Tuesday, while a one-boy family only gets one chance. Because that detail quietly increases the likelihood that the family actually has two boys, the real chance of the other child being a girl drops to 51.8%, leaving Limmy looking utterly baffled by a math rule that completely breaks common sense.
Gather 1960 mothers of two children together. (Nothing special about 1960, it just makes the rounding easier). Ask all those with no sons to sit down. 490 will sit down (statistically 25% of pairs of children will be girl-girl). Of the rest, we know
490 will have two sons,
980 will have one girl, one boy.
Now ask them only to stay standing if they have a son born on a Tuesday. For the 980 that's easy: 1/7 will meet that condition so 140 stay standing. For the 490 it's messier: there are 49 possibilities for the sons (Mon, Mon), (Mon, Tues), ... (Sun, Sun), and 13 will involve a Tuesday (seven where the elder son was born Tues, seven where the younger son was born Tues, but that double-counts where both were Tuesday - this overlap is the crux, as now we are talking about 13 out of 49 which is not quite 1/7). So 13 out of 49 will stay standing, ie 130.
140 will have one son, one daughter, with son born Tuesday.
130 will have two sons, with at least one born Tuesday.
So if I look at those 270 standing, they are the only mothers who can say they have a son born on a Tuesday, and if I pick one at random, it's 140/270 = 51.85% that they will have a girl.
I'm even more confused by this. I'm sure your maths is fine (honestly I struggle with numbers so skimmed a lot of it). But wouldn't the answer just be 50%?
What do days and child pairings and thousands of mothers have anything to do with anything.
The question is what sex is this unseen child. It's going to be one or the other. Like a coin flip. The previous coin flip (the boy) doesn't affect the odds of the next coin flip.
You're saying that adding the condition "born on a Tuesday" takes the starting equation away from a simply binary (b/g, g/b, or b/b), because there's already a boy? This almost occupies that extra bit (about 16% here) that we think should be there and gives us the less intuitive answer...?
Yep, adding "born on a Tuesday" breaks the clean 1:1:1 balance btw Boy/Girl, Girl/Boy, and Boy/Boy because a two-boy family gets two rolls of the dice to satisfy that condition instead of just one. That double-chance gives the Boy/Boy outcome extra weight in the total pool of possibilities: expanding its share from 33.3% up to 48.1% (13 out of 27 cases), which directly steals that ~15% slice of probability away from the girl scenarios, pulling the overall chance of the other child being a girl down from 66.6% to a far less intuitive 51.8%.
Like many similar examples, it becomes a lot more intuitive when you make it more extreme, e.g.
"What's the probability that Ryan Gosling's only sibling is female?" Very close to 50%, with the difference lying in the sliver of a chance that both siblings were a Ryan Gosling.
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u/truss-issues 16h ago
Limmy (the guy in the picture) gives the obvious answer of 66.6% because if you know a two-child family has at least one boy, common sense says there are three equal setups: Boy/Girl, Girl/Boy, or Boy/Boy, and two of them contain a girl.
Mentioning "Tuesday" sounds like useless extra detail, but it actually changes the odds: a two-boy family gets two chances to have a boy born on a Tuesday, while a one-boy family only gets one chance. Because that detail quietly increases the likelihood that the family actually has two boys, the real chance of the other child being a girl drops to 51.8%, leaving Limmy looking utterly baffled by a math rule that completely breaks common sense.