61–70 of 152 posts
As the currently top voted comment does not get it, I try to intuitively explain the paradox. No, it is not ambiguity of language. It says formally: I have two children. There exists a child of mine who is (boy and born on tuesday). And yes, the probability of the other child being a boy is 13/27. To understand this, try a more extreme case: When a child is born we generate a random number: rnd(1billion) Now the man…
Earlier quoted context omitted.
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 ou…
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.
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".
Earlier quoted context omitted.
No I don't think that's true. Bear with me :-) 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 b…
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.
Earlier quoted context omitted.
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.
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 ----…
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.
Earlier quoted context omitted.
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.
ajanuary explains the duplicates problem nicely here: http://news.ycombinator.com/item?id=3290185 so I won't double up. 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.
ANYONE WHO SAYS THE ANSWER IS OTHER THAN 1/2: Let's simplify the question: If a man says "I have two children, one was born on a Tuesday," what is the probability that they are both born on a Tuesday? Is the answer to this 0 or 1/7 in your opinion? (Or something different).
Child 1 born Mon, Tue, Wed, Thu, Fri, Sat, Sun
Child 2 born Mon, Tue, Wed, Thu, Fri, Sat, Sun
We know that one of the children was born on a Tuesday, but we haven't said whether it's Child 1 or Child 2 (and that lack of information is key to understanding the original problem), so our unknown child could also be either Child 1 or Child 2.
With that in mind, 'unknown child' has 14 possible options except that we know one of those options (our known Tuesday child) is already taken - we don't know whether it's Child 1 or Child 2 but that doesn't matter, we just know that one of them is already taken. That leaves 13 possibilities for unknown child, only one of which is 'born on Tuesday' (since we've removed the other 'born on Tuesday' option), so I would say the answer is 1/13.
Earlier quoted context omitted.
Since the constraint simply says that at least one child is a boy, we don't need to distinguish between two types of BB. This is a curious misconception, by the way. The typical failure mode I see on these questions is people having difficulty accepting BG and GB as different possibilities.
If we are determined to look at order of birth (ie. have a BG and GB) then we should consider all cases by order of birth, so where 'B' represents the boy we know and 'b' or 'g' represents a child we dont, and the first character represents the first child, and the second the second childe - we have: Bb, bB, Bg, gB 50%.
However, you're doing this on an already selected sample by discounting all the girl-girl pairs. The selection rule in your logic then becomes a two-step "If you have at least one boy, then pick a child at random, then if it's a boy, say: [blablabla]"
Given that, your analysis is correct.
But, seeing as it is a wildly different selection model than what everyone else seem to work with, you should be explicit about it.
Earlier quoted context omitted.
They are independent, but you get information about both ("One of them is a boy."), therefore p=1/3. If you say "the older is a boy" then you have a statement about one son and no information whatsoever about the younger one, therefore p=1/2.
"One of them is a boy." is a completely different scenario and no one is arguing about it. (BG, GB, BB, GG - by saying one is a boy you exclude GG leaving 1/3.) In this case we are asking if the other is a boy NOT if the other was born on Tuesday.
You are asking, why the information "one is born on Tuesday" says anything about the gender or birthday of the other, right?
More specificaly, this:
> Let us first assume that it is the older child who was a son born on a Tuesday. In this case the second child could be either of two sexes, and could have been born on any of seven days of the week, for a total of 14 possibilities.
> Now let's suppose it is the younger child who was a son born on a Tuesday. Then the older child could, again, be either of two sexes and could have been born on any of seven days of the week, again providing 14 possibilities. Added to our original 14 that would seem to give 28 possibilities.
> But be careful! One possibility got counted twice. Specifically, the one where both children are boys born on Tuesdays. So really there are only 27 possibilities. And since 13 of them involve the second child being a boy, the probability would be 13/27.
Maybe it helps, if you first imagine all 196 (2 * 2 * 7 * 7, think of four 7-by-7 tables [draw them, it helps a lot ;)]) possibilities. Eliminate 49 of them where both are girls (one of the four tables) and you have 147 possibilities/cells left. Then, in each of the two BG and GB tables leave only one row/column, eg. eliminating another 2 * 6 * 7=84 possibilities.
Now comes the interesting/tricky part: The last case/table, where both are boys contains only 13 possible cases (instead of 2 * 7)! Imagine a 7-by-7 table, where each row and column corresponds to a day. Mark all the cells which are not in a Tuesday-row or a Tuesday-column, eg. eliminate all posibilities where none of the two boys are born on a Tuesday. This eliminates another 49-13=36 cases.
So, we have a total of 196-49-84-36=27 cases, 13 of which "the other is a boy" and 3/27=1/9 cases where "the other is born on a tuesday".
I hope this makes somewhat sense. Once I've drawn all the possibilities and marked all the impossible ones, it became a lot clearer.
As the currently top voted comment does not get it, I try to intuitively explain the paradox. No, it is not ambiguity of language. It says formally: I have two children. There exists a child of mine who is (boy and born on tuesday). And yes, the probability of the other child being a boy is 13/27. To understand this, try a more extreme case: When a child is born we generate a random number: rnd(1billion) Now the man…
This is a big call to make, seeing as English is not your native language. As a native English speaker, I find it very ambiguous as to which set of probabilities I should be counting.