Note that we haven't said which child is 1 and which is 2, and in fact we don't know because we haven't been told - this is the important bit.
So, for our unknown child to be a boy, it must be one of the following 14 permutations:
unknown child is Child 1, boy, born Mon-Sun
unknown child is Child 2, boy, born Mon-Sun
but one of those permutations is already taken by our known Tuesday boy, so we have to remove one of the Tuesday permutations leaving 13 possible outcomes out of 27. Note that one of those 13 outcomes is 'unknown child is son born on Tuesday' - it's perfectly valid to have both sons born on a Tuesday.
Note also that we can't say which permutation we're removing until we know which is Child 1 and which is Child 2, just that one of them is taken.
If the original statement were phrased as '... the older child is a son born on a Tuesday' then you would have a constraint on which was Child 1 and which was Child 2 and then the probability would be 1/2 as expected because you would know up front that you were entirely discounting, say, Child 1, AND that the removed Tuesday permutation also belonged to Child 1.
First, saying Mon-Sun is confusing almost everyone since most people start the week on Sun, not Mon. (Even in Europe Christians start the week on Sunday.)
In any case, as I replied here http://news.ycombinator.com/item?id=3290118 you can not remove the duplication! It's two different situations, even though they may appear the same.
I've spent the last hour and a half wrestling with the same clash between maths and intuition that you're experiencing so I understand your frustration - 13/27 really is the right answer though.
Look, I understand your intuition wants you to remove duplicates "They are the same!". But you should resist, because doing that gives incorrect results.
On the contrary - I'm saying that 'removing the duplicates' is counterintuitive and that's why you're having trouble with it, the same as I did.
Besides: please re-read my original post - I didn't mention anything about removing duplicates, and in fact chose to explain the problem in a different way precisely because the duplicate removal thing was so difficult for me to grasp sufficiently to be able to explain it.
Ask yourself this: where does the original 28 come from? One of those 28 permutations appears to be 'Child 1 is Tuesday boy' and 'Child 1 is Tuesday girl'. If you can answer that then you should be able to understand the whole thing.
There are 14 possible permutations for a child
Mon-Sun Girl
Mon-Sun Boy
which for two children gives a total of 28:
Child 1 is Boy, born Mon-Sun
Child 1 is Girl, born Mon-Sun
Child 2 is Boy, born Mon-Sun
Child 2 is Girl, born Mon-Sun
Note that we haven't said which child is 1 and which is 2, and in fact we don't know because we haven't been told - this is the important bit.
So, for our unknown child to be a boy, it must be one of the following 14 permutations:
unknown child is Child 1, boy, born Mon-Sun
unknown child is Child 2, boy, born Mon-Sun
but one of those permutations is already taken by our known Tuesday boy, so we have to remove one of the Tuesday permutations leaving 13 possible outcomes out of 27. Note that one of those 13 outcomes is 'unknown child is son born on Tuesday' - it's perfectly valid to have both sons born on a Tuesday.
Note also that we can't say which permutation we're removing until we know which is Child 1 and which is Child 2, just that one of them is taken.
If the original statement were phrased as '... the older child is a son born on a Tuesday' then you would have a constraint on which was Child 1 and which was Child 2 and then the probability would be 1/2 as expected because you would know up front that you were entirely discounting, say, Child 1, AND that the removed Tuesday permutation also belonged to Child 1.
But I stand to be corrected!