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
The question is notoriously ambiguous.
If i have a son, and i say to you “hey this is my son timmy. I also have another child. Do you think it’s a boy or a girl?” The probability you guess right is 1/2. You don’t know which brother is timmy, but *i do*. It is my family, and i gave you a precise information, you just don’t know it. If i know timmy is the oldest i’m actually asking the probability my family is boy-girl or boy-boy. I’m not throwing girl-boy into the mix because i know timmy is my older son, despite i haven’t told you. Conversely, if i know timmy is the youngest i’m just asking girl-boy vs boy-boy. You don’t know which question i’m asking you, but it doesn’t matter because the answer is 1/2 to both. I’m basically just asking you to guess the gender of a person you don’t know, in a fancy way. The answer is in fact 1/2 but this is NOT what your question is asking.
On the other hand your question is more like this: consider all the families in your hometown with exactly two children. I choose one at random and go to their house. I only see one boy – don’t know if it’s the older, younger, and haven’t seen the other child. Based on that, what’s the probability i chose a family with two boys? Well, there are 4 ways a 2-children family can be: GG, BB, BG or GB. Suppose they are all equally likely. Can’t be GG because i saw the boy, but i don’t know the family, so it’s equally likely i saw the older sibling of a BG family, the younger of a GB family, or either one of a BB family. Putting the probability of BB at 1/3.
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