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In this Probability Theory course, the question

A girl I met told me she has one sibling. What is the probability that her sibling is a boy?

is asked, with the answer stated as $\frac{2}{3}$. I'm confident that the correct answer considering the wording is actually $\frac{1}{2}$, but this may have been a small typo in the question. Then the followup claim

Somebody told me that one of the children is a girl, and that girl was born on a Saturday. Now the probability of the other child being a boy is not $50\%$; it's not even $\frac{2}{3}$.

is made, which gets restated as

One of my $2$ children is a girl; she was born on a Saturday. What is the probability that the other child is a boy?

If the first quote was intended to be worded similarly to the latter two, then $\frac{2}{3}$ would have been the answer to the first question, since "one of my $2$ children is a girl" selectively refers to whichever child is a girl, in the event there is exactly $1$, whereas meeting a girl gives no indication of this intelligent selection.

However I can't figure out how one arrives an answer other than $\frac{2}{3}$ or $\frac{1}{2}$ once the Saturday bit is added. Someone in the comment section guessed $\frac{14}{27}$. How does this new information, on either the "one of the children" or "the child I met" wording, produce a new answer? If (as I suspect may be the case) it actually doesn't, what is the argument for becoming falsely convinced of the wrong answer, and how is it rebutted?

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Suppose you meet a girl with one sibling. The probability that the sibling is a boy is $\frac12$ because the gender of the sibling is independent of the girl you know about.

Now someone tells you they have two children and one of them is girl. There are three equally likely possibilities $GB, BG, GG$ and two of them involve the other child being a boy, so the probability of a boy is $\frac23$.

The point here is that the girl you know about is not distinguished from the sibling you do not know about. If you were told that the elder child was a girl, then you would know nothing about the other one, and the probability of the other child being a boy would be $\frac12$ again. In the first problem you met the girl which distinguished her from her sibling, so the probability of the sibling being a boy was $\frac12$.

Finally, if you are told that one of two siblings is a girl born on a Saturday, then you are between the two cases. The sibling is not completely distinguished from the girl you know about, because they may both be born on a Saturday. If you are told that the other child was not born on a Saturday, then the probability of the other child being a boy is $\frac12$ because the genders of the Saturday child and the non-Saturday child are independent.

Conversely if you were told that both children were born on a Saturday, then the probability that the other child is a boy would be $\frac23$.

As you are not told either way if the other child is born on a Saturday, you expect an answer between $\frac12$ and $\frac23$, but much closer to $\frac12$, as the other child is most likely to not be born on a Saturday.

That is why an answer of $14/27$ should not surprise you. Now for the calculation:

There are 14 possibilities for the first child (gender and day of week) and 14 possibilities for the second. However one of them must be a girl born on a Saturday, so there are 27 valid combinations. 14 of them have the sibling being a boy so the probability of a boy is $\frac{14}{27}$.

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