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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 $P(ALOG|BG) = \frac{1}{2}$?

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  • $\begingroup$ There’s no paradox there per se. The problem illustrates that people’s naive intuition about conditional probability is often wrong. $\endgroup$
    – amd
    Commented Apr 4, 2020 at 20:03

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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.

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    $\begingroup$ The thing is, if a randomly chosen one of two children turns out to be a girl, to phrase that conditional information as ALOG (“at least one girl”) is a crime against clear and honest communication. 😁 $\endgroup$
    – Ned
    Commented Apr 4, 2020 at 20:28
  • $\begingroup$ @Ned: Well, that's sort of looking at it in the rearview mirror. Martin Gardner wrote this column; people rightly complained that the information contained in the statement "At least one of them is a boy" was ambiguous as long as we don't know how this statement came to be made; and now we're analyzing one way of defining how this statement could come to be made. I certainly agree with you that, if this were what was meant, "at least one girl" would not be the best way to express it. But that's not the question; the question is how "at least one girl" is to be interpreted :-) $\endgroup$
    – joriki
    Commented Apr 4, 2020 at 20:32
  • $\begingroup$ I'd go further, the chart as shown is guaranteed to be confusing to anyone who doesn't already understand the whole situation, since ALOG is a good phrasing of the other case. It would be better presented with a different acronym, like RCWG "Random Child Was a Girl" or whatever 😁. $\endgroup$
    – Ned
    Commented Apr 4, 2020 at 21:27

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