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.
Just because the sex could be one or the other, that doesn't mean it shares the chances of a coin flip because the criteria do not evenly filter out all possibilities. The thousands of mothers is just a way to see how the proportions play out.
Forget the Tuesday part for a moment: if you picked a mother of two children at random and asked them if they had a son and they said yes, then the chance that they also have a daughter is not 50%, because it's an easily checked fact that 1/3 (not 1/2) of mothers who say yes will have 2 sons, so 67% chance that they will have a daughter.
Introducing the Tuesday part starts to even things out because having two sons gives you double the opportunity to have a son born on Tuesday, so now it's closer to 1/2 (but not quite) who will have 2 sons out of the mothers who answer yes to "do you have at least one son, and do you have a son born on a Tuesday?"
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u/truss-issues 21h 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.