# The probability that the second child of a couple whose older child is a boy is also a boy

Someone is meeting a couple which has two children. One of the children, which is a boy, joins them. What is the probability that the OTHER child is also a boy if we know the OTHER child is younger.

I think the answer must be $\frac 12$. But what is wrong with the following:

Since we know one child is already a boy, the sample space, in order of birth, would be $A = \{ (boy,girl) , (boy,boy) , (girl,boy) \}$. And since we want the younger one to be a boy then the probability would be $\frac 13$.

• Its okay to down vote ploblems, but it would be nice to leave a note or write a reason. Commented Nov 15, 2015 at 10:51
• I would wish too one day SE would implement punishment for those who downvote without comments or with nonsense comments. Commented Jun 15, 2023 at 8:35

Considering that the first element in the set is the older child: (girl,boy) will not exist

A={(boy,girl),(boy,boy)}

You can not consider

A={(boy,girl),(boy,boy),(girl,boy)}


because you already know that the elder child is a boy, so the chances for the second child also to be a boy is 1/2.

A={(boy,girl),(boy,boy)}


that is why

P(x)=1/2


x is the probability the other child is also a boy.