I am reading https://en.wikipedia.org/wiki/Boy_or_Girl_paradox and I am confused about the following table they showed.
In the second row, where you are considering a family of 1 girl and 1 boy, and you are asked what is $P(ALOG|BG)$ where $ALOG$ is "at least one girl" and $BG$ is a family with a boy and a girl. If you are given this family, then isn't the probability of having at least 1 girl $P(ALOG|BG) = 1$ and not $P(ALOG|BG) = \frac{1}{2}$?
The article isn’t all that clear in explaining this, but it’s correct.
The probabilities in the table are not, as you seem to be assuming, the probabilities of the family having at least one girl/boy, but the probablities of the statement “The family has at least one girl/boy” being made about the family. The table is in the part of the article that discusses one of two possible interpretations of the problem statement, namely the one in which a family with two children is selected, a child in that family is selected at random, and a true statement is made based on its gender. In this scenario, the probability that the statement “The family has at least one girl” is made if this family is selected is indeed $\frac12$, since it is made if the girl is uniformly randomly selected, with probability $\frac12$.
It is perhaps also worth making explicit that the table assumes the approximations that each child is either a boy or a girl and that these possibilities are equally likely.