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
Humans are born male or female with probability 50% (for the purposes of simple probability problems).
So in a scenario where you’re trying to guess a human’s sex with no additional information, the probability would be 50%. For example: “This is Jimmy. He has 1 sibling. Do you think he has a brother or a sister?” Here, his sibling is a human, therefore there is 50% probability they are either sex.
This scenario is like if you toss your favourite pet coins, Athena and Barbara, and look at Athena but not Barbara. Barbara, like all coins, is 50/50.
The interpretation that results in 1/3 is if you read the words “I have two children. At least one of them is a boy. What is the probability I have two sons?” and say okay, I don’t have any information about any human. There is only a whole circumstance including two humans: at least one is a boy.
It’s like if you’re on holiday, and you ask your friend to toss your coins and tell you whether there are any heads. If they tell you yes, what do you think both of them are? Note I can’t say “the other one” because there is *no* “*the other one*” — *you don’t know* which one is heads. The three different equally likely possibilities in the circumstances as described are HT TH HH (where I’m writing Athena on the left and Barbara on the right).
Whether this interpretation is a reasonable way to read these words is something that people disagree about.
I hope this helps!
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