And yet it changes the odd.
I like this formulation: It gets to the right information, but in a way that looks suitably close to natural (or at least, natural enough for a riddle). It also provides a framework for saying why the information matters:
Suppose that, instead of just one eccentric millionaire, this becomes a fad among millionaires. Every millionaire randomly chooses a gender, and then randomly chooses a day, and then out of all of the families who apply, randomly chooses one of those families. If you’re a parent of two children, in the world where this fad has taken hold, you’ll be unhappy if it happens that both of your children have the same gender and day, because that will decrease your chances of being eligible for the next millionaire’s lottery.
This is the problem when you get a bunch of smarty-pants together; they tend to overthink things. Let’s go back and look at the original problem/question — quote:
I have two children. One is a girl born on a Thursday.
What is the probability both of my children are girls?
Note the question is NOT “what is the possibility my other child is a girl?” It is also not “what is the possibility that my other child, whatever it might be, was also born on Thursday?” It is not even “what is the possibility of my second child, when it is born, also being a girl?”
No, it is “what is the probability both of my children are girls?”
And since the two children have already been born, it could have only happened according to one of the four possibilities below:
1 girl, 1 girl
1 girl, 1 boy
1 boy, 1 girl
1 boy, 1 boy
Therefore, the probability of the questioner having two girls is only one of the four possible combinations, meaning it is 1 out of 4 or 25%. The bit about the one girl being born on Thursday is just obfuscation.
-“BB”-
Bicycle_Bill, it is different.
If I said two children and at least one is a girl AND you then asked me what day at least one girl was born and I said Thursday, then nothing changes. But if I tell you the day without your asking then the g/g possibility is 15% or so higher.
For every problem there is a solution that is simple, neat—and wrong.
Or think things to the appropriate level to get the right answer, rather than definitively declaring the wrong one, as you have done here.
Given that one of the pieces of information is that they have one girl, it seems as though one of the possibilities that you have listed is not, in fact, possible.
The answer to the question as stated in the OP cannot be 25%, as you have so declared, as that would involve the possibility that both the children are girls, and that has already been ruled out. As the question is stated in the OP, the answer would be 50%, as you now are only really asking about the gender of one child.
As has been discussed, depending on how the problem is worded, that changes things. If worded correctly, the date of birth can actually be information that changes the probability.
I asked my brother-in-law the statistics professor – he says the answer is 13/27.
So is there an actual answer, or was this an exercise in speculating?
Word problems rely on varying levels of interpretation of language, so it’s hard to be definitive. But the intended answer was 13/27.
Ha Ha-- hahahahaha!
I suck at math generally, but I can do stats and set theory, for some reason.
AIUI this is still wrong. If someone asks you if at least one of your girls was born on Thursday, and you say yes, that changes the probability. But if you get to pick what day to tell us, then it doesn’t necessarily change the probability. Someone else needs to specify the sex and the day, not you.
I can’t believe that I’m the only person who Googled this, and found a quite logical explanation of the answer:
Also, @RivkahChaya, your explanation is wrong, in that you state that ‘Girl is child A, born on any day, while boy is born on Thurs, or girl is child b, born any day, while boy is born on Thursday.’ If the other child is a boy, then the girl child is definitely born on Thursday.
Well, I think I was speaking about all the possibilities, not the limited ones constrained by the conditions. The denominator is all possible outcomes, and the numerator is the potential one as constrained by the conditions. IIRC. I’m not actually looking back at what I wrote because I can’t find it.