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
If at least one child is a son then there are three possible outcomes:
1) Oldest is a boy, youngest is a girl
2) Oldest is a boy, youngest is a boy
3) Oldest is a girl, youngest is a boy
Equal odds of any of these being true, so 1/3
If you list out the different possible outcomes, you can see it more easily. You could have:
BB
BG
GB
GG
Since you know at least one of them is a boy, you throw out the two girls outcome. Thus, there are three outcomes possible, and only one of them is two boys.
There are three equally likely ways he could possibly have two kids with at least one boy:
1. Child 1 is a boy, child 2 is a girl
2. Child 1 is a girl, child 2 is a boy
3. Child 1 is a boy, child 2 is a boy.
One of those three possibilities is two boys, making that a 1/3 probability.
There are 4 possibilities
Boy Girl
Girl Boy
Boy Boy
Girl Girl
Each one is equally likely (Boy Boy is a 1 in 4 chance)
Knowing at least one of them is a boy eliminates the Girl Girl possibility
Now we have 3 possibilities
Boy Girl
Girl Boy
Boy Boy
Only one of those 3 possibilities has two boys, so the odds are 1 in 3
For 2 children, the possible options are:
* B, G
* B, B
* G, G
* G, B
Only three of those possibilities have a boy in them somewhere, and “B, B” is a single one of those three possibilities. So it’s 1/3.
In addition to what others have said, you just need to focus more specifically on the wording of the question.
You’re being the asked the probability of the man having “two sons”, not the probability of the other child being a boy.
Two kids; one boy. So we only care about the other child; 50/50 odds boy or girl. So it’s common to say 1/2 odds that they are both boys.
Except we are actually looking at a variant of the Monty Hall problem.
Initially, looking at two kids; independent odds of gender, so four possibilities:
boy, boy;
boy, girl;
girl, boy;
girl, girl;
So 1/4 odds that you got “boy, boy”.
At least one boy; so remove the girl, girl option.
So 1/3 odds for “boy, boy” from the remaining options.
Unless there’s more in the second question, day of week wouldn’t matter at all.
Something worth mentioning that I haven’t seen yet, the reason there are four options is because the actual real-life probability of having two boys or two girls is only 25% (flipping a coin twice). Having one of each is 50%. So it is important to separate these options out when calculating the probabilities.
So, everybody claims the order is important when people who “dont get it” ask for clarification.
Lets say mum is pregnant with twins. They cut her open, both twins pop out at the same time. Whats the difference between BG and GB?
If you aim to have at least one boy, you have two chances to get that right. For two boys, you only have one chance to get that right.
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