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Ok, so in my textbook I have the boy or girl paradox. I managed to solve it, but there is a slight modification of the paradox that goes like this:

You are invited to a family that has two children. A boy opens up the door for you. Suddenly, you hear a baby cry. What's the probability the second child will be a boy?

I am posting this question just to be sure- the baby has no impact on the probability, right?

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    But this just seems to be a question of phrasing. Are we to assume that the boy is one of the two children and that "the second child" simply refers to the one that isn't standing in front of you? If so, the answer would appear to be $\frac 12$, under the usual assumptions. Or are we to read the question differently? – 2017-01-22
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    For example; one might take "the second child" to mean the youngest of the two, in which case one might also be tempted to guess that this would be the baby (so the answer would again be $\frac 12$). – 2017-01-22
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    @lulu The way I understand it, it that yes indeed, the boy is to be assumed the first child, and the crying baby the second child. I had a question in my textbook before this one, where the phrasing was exactly the same, except that there was no crying baby. I don't see what the baby has to do here. – 2017-01-22
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    @lulu I doubt that it would mean that the second child would be necessarily the youngest, but then again, there is no precision in my textbook. – 2017-01-22
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    In normal speech, "second child" certainly means second in age. And, if we read it that way here, then we can be fairly certain that the baby is the second child, not the first. But...this isn't really a math problem. Just some difficulty translating from imprecise English to math. – 2017-01-22

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That would be $\frac{1}{2}$. So yes, it's different from the 'standard' 2 children and one is a boy' problem, since in this case you know that one specific child is a boy, whereas in the 'standard' version you are merely told that 'at least one of the children is a boy'.

The mention of the baby makes no difference .... Even if you know that the baby is the second child (which of course you don't really know ... Then again, do you know the boy is one of the children? ...)

Anyway: you know that besides this one child being a boy there is one other child. And given no further information about the gender of this second child, the probability of it being a boy is $\frac{1}{2}$

If you delete the sentence about the baby from this problem description, the probability is still $\frac{1}{2}$. Also, if it turned out that the baby was not the second child, the probability is still $\frac{1}{2}$.