I was asked the question “a man states he has two children, and at least one of them are boys, what are the odds that the man has two boys?” I’ve been told the answer is 1/3, but I can’t wrap my head around it. Additionally, there is another version of the problem that states he has at least one boy born on a Tuesday. How does that change the odds? Why?
Edited to add (so people don’t have to sort through replies): the answer is 1/3 because “at least one boy” is accounting for B/G & G/B. The girl can be the first or second child. You can move the odds to 50/50 by rewording the question to “my first of two children is a boy, what are the odds the other child is a boy”
In: Mathematics
It is a bit more complex than your edit because the question is a bit deceptive. Most means of knowing that one of a man’s two children is a boy would make it 50/50 on the other being a boy. For example, meeting one of the boys, the man mentioning his son did something, etc. One of the few ways to get this scenario is to ask the man with two children if he has any sons, and he says yes. What are the odds he has no daughters? 1/3. Another way is to know a king has two children, and he has an heir, and heirs can only be male. The odds he has two sons is 1/3. Or, you meet the heir to a king, and he has 1 sibling. The odds are 1/3 that sibling is a boy.
Most natural ways of finding out someone has a son makes it such that there is a 50/50 chance of them having two sons if they have two children because them mentioning a son sort of switched the perspective to being you know a boy has a sibling. What are the odds that sibling is a girl? Well, 50% of boys have a sister and 50% have a brother even though only 2/3 of families with a boy have a boy and a girl while only 1/3 are boy boy.
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