"I have two children and one is a son born on a Tuesday."
"I have two children and only one of them is a son
born on a Tuesday."
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!
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.
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.
(edited for clarity)
Once you include that possibility it's back to 14/28.
When considering the case that the younger son was the one born on a Tuesday, that gives 14/28 possibilities. One of those 14 is the case that both were born on a Tuesday.
But woops, we've already covered the case that both were born on a Tuesday in our first count. Removing the duplication gives the 13/27.
You then combine those two probabilities.
The 13 wasn't arrived at by excluding the possibility that the second child was a boy on Tuesday, it was arrived at by discounting the case that they were both born on Tuesday as it had already been covered in the previous sub-calculation.
If you have two kids there are 4 possibilities, not 3: BG, BG, BB, GG
BG seems to be the same as GB, except that it's not. And it's not in this case either.
Assuming Child 1 is a Boy:
Child 2 can be: Boy, Girl
Assuming Child 1 is a Girl:
Child 2 can be: Boy, Girl
Assuming Child 2 is a Boy:
Child 1 can be: Boy, Girl
Assuming Child 2 is a Girl:
Child 1 can be: Boy, Girl
Combining those gives us the following combinations:
Child 1 | Child 2
---------+---------
Boy | Boy
Boy | Girl
Girl | Boy
Girl | Girl
Boy | Boy
Girl | Boy
Boy | Girl
Girl | Girl
Clearly there is duplication in there we need to remove any exact duplications before we get to the 4 possibilities you listed.
Now looking at the problem in the article again, but using a 2 day week for brevity:
Assuming Child 1 is a Boy on Tues:
Child 2 can be: Boy on Mon, Boy on Tues, Girl on Mon, Girl on Tues
Assuming Child 2 is a Boy on Tues:
Child 1 can be: Boy on Mon, Boy on Tues, Girl on Mon, Girl on Tues
Combining those gives us the following combinations:
Child 1 | Child 2
--------------+-------------
Boy on Tues | Boy on Mon
Boy on Tues | Boy on Tues
Boy on Tues | Girl on Mon
Boy on Tues | Girl on Tues
Boy on Mon | Boy on Tues
Boy on Tues | Boy on Tues
Girl on Mon | Boy on Tues
Girl on Tues | Boy on Tues
In exactly the same way, removing exact duplications gives us 3/7.
There is no ordering over the Tuesdays. "The older child being born on Tuesday and the younger child being born on Tuesday" is exactly the same as "The younger child being born on Tuesday and the older child being born on Tuesday". Just like "The older child being a boy and the younger child being a boy" is the same as "The younger child being a boy and the older child being a boy".
Also, why are you numbering the kids as child 1/2? There is no such distinction made. If you changed your list so that the fixed child is always listed first, and removed duplicates you would have 4 possibilities.
If you're not meant to remove the duplicates, then why doesn't your enumeration of two gendered children run: BB, BG, GB, GG, BB, GB, BG, GG?
I number them merely to distinguish them. The "a child has 50% chance of being a boy, and 50% chance of being a girl" applies to a single child, so you need to enumerate their states independently. To do that you need to be able to distinguish between them.
It is this fact that each child's possible states should be treated independently that means you can't combine BG and GB, not any aversion to removing duplication. Because they are independent entities BG represents one - nominally called child 1 - is a B while the other - nominally called child 2 - is a girl. GB represents one - nominally called child 1 - is a G while the other - nominally called child 2 - is a B.
If the list were changes so that the fixed child is always in the list we wouldn't be enumerating all the possibilities for each child and then combining them. That would be falling into exactly the same mistake you are keen to avoid.
() Where BT is "Boy born on a Tuesday"
BG is only not the same as GB if there is some other information available - which was born first, what their names are, hair colour, etc., because then you'd be saying something like
Boy born first, Girl born second
Girl born first, Boy born second
and those are two distinct possibilities. The point is that they are only distinct if you have this extra information, which we don't. We have no way of differentiating between the two children except for gender, and therefore BG and GB both just say 'one male child and one female child'. You might assume that the order of the two letters specifies the order in which the children were born, but that's a false assumption because it's not stated anywhere.
If you include the order, there are four possibilities: a) Boy first, Boy second, b) Boy first, Girl second, c) Girl first, Boy second, d) Girl first, Girl second. If you don't, there are three possibilities: a) two boys, b) two girls, c) one boy and one girl. The unspecified information about order is the subtle but important difference.
It is not order that is important, but considering each child as a distinct entity.
The 50% chance of being a boy and 50% chance of being a girl applies to a single independent child. When enumerating the possible combinations we need to first enumerate the possibilities for each child, and then combine these two enumerations into our overall enumeration.
To help us distinguish between the two children, let's call one Sam and the other Alex. There's no ordering over them, it's just to help us tell which one we're talking about.
Sam can be: Boy, Girl
Alex can be: Boy, Girl
Combining those gives us:
Sam is a Boy, Alex is a Boy
Sam is a Boy, Alex is a Girl
Sam is a Girl, Alex is a Boy
Sam is a Girl, Alex is a Girl
Or, to use a shorter notation, BB, BG, GB, GG.
Using oldest and youngest is just another handy way of distinguishing between the two children. Even if they were nameless, faceless children with no distinguishing characteristics other than gender we still need to consider then separately, each as their own entity with their own enumerations of possible genders.
Where the parent commenter is wrong is saying that (correctly) considering GB and BG as distinct combinations is the same as considering "Oldest Boy born on Tuesday and Youngest Boy born on Tuesday" distinct from "Youngest Boy born on Tuesday and Oldest Boy born on Tuesday". They are not. When you stop thinking about ordering and instead think about the enumerated states of each entity (child) involved it becomes clear both are saying the same information. They are not distinct, but the same combination phrased differently.
The parent commenter is basically saying BB should be distinct from BB just because the first one talks about Sam first and the second one talks about Alex first. This is clearly wrong.
"I have a son and a daughter"
is the same as
"I have a daughter and a son"
unless you qualify the statement with further information - names (Alex and Sam from your post) would be an example of that further information. My feeling was that ars is implicitly 'filled in the blanks' somewhere, treating the two children as distinct when in fact they have to be interchangeable for the purposes of the original (13/27) calculation.
It's the lack (or not) of such details that alters how the probability is calculated.
I admit though that I'm still chasing myself in circles trying to understand the whole thing so take this reply with a pinch of salt ;-)
If it were about the phrasing you could assume the other child is a daughter - if he were to continue ... "The other is also a son, born on a Friday" it would just sound strange.
Perhaps it is all down to his turn of phrase :)
edit: accidental downvote, sorry.
edit2: but see my top-level reply to OP.