A family has two children. One child is a girl. What is the probability that the other child is a boy? [duplicate]

You should not have omitted the BG!

The sample space is (and needs to be) $\{BB, BG, GB, GG\}.$

We know one child is a girl, so that rules out BB. That leaves us with a sample space of BG, GB, GG

In which case the probability that the second child is a boy is $\dfrac 23$.

Each of the four outcomes has a probability of $\frac 14$. BG: "Having a boy, and then a girl" is a different outcome than $GB:$ having a girl, and then a boy.

To omit one of the boy-girl/girl-boy pairs leaves a sample space of three, with each outcome having probability of 13, which is not correct. Having a boy-girl pair is twice as likely as having two boys, and twice as likely as having 2 girls, and we can only obtain this by counting all four outcomes as distinct, indeed, distinguishable.