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When considering the case that the older 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 T

When considering the case that the older 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 Tuesday. 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.

ars 15y ago
No, you do not remove the duplication. 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.
- ajanuary 15y ago
  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"
- danparsonson 15y ago
  I think I've spotted where your misunderstanding is. 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.
  - ajanuary 15y ago
    No, they are correct in saying that GB and BG are distinct, even with no other information. 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.

ars 15y ago

No, you do not remove the duplication. 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.

ajanuary 15y ago
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"
danparsonson 15y ago
I think I've spotted where your misunderstanding is. 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.
- ajanuary 15y ago
  No, they are correct in saying that GB and BG are distinct, even with no other information. 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.