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Looking at your simple example of genders using the same process the article uses to enumerate the combination of gender and days: Assuming Child 1 is a Boy:

Looking at your simple example of genders using the same process the article uses to enumerate the combination of gender and days: 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".

ars 15y ago
Except you are not actually supposed to remove the duplicates!! It is a very common mistake, but it's simply incorrect, it leads to incorrect results. 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.
- ajanuary 15y ago
  The removal of duplication and the distinguishing between GB and BG are two completely different things. 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.
- ajanuary 15y ago
  I would be interested in your reasoning why G,B and B,G can (rightly) be considered distinct, and B,B and B,B can (rightly) be considered the same enumeration and thus one discounted, but BT,BT and BT,BT () should still be considered distinct when it is clearly the same situation as B,B? () Where BT is "Boy born on a Tuesday"

ars 15y ago

Except you are not actually supposed to remove the duplicates!! It is a very common mistake, but it's simply incorrect, it leads to incorrect results. 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.

ajanuary 15y ago
The removal of duplication and the distinguishing between GB and BG are two completely different things. 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.
ajanuary 15y ago
I would be interested in your reasoning why G,B and B,G can (rightly) be considered distinct, and B,B and B,B can (rightly) be considered the same enumeration and thus one discounted, but BT,BT and BT,BT () should still be considered distinct when it is clearly the same situation as B,B? () Where BT is "Boy born on a Tuesday"