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jackWhat all this Leno/Conan/Late Night Gubbins is about: a primer for friends in the UK
http://littleredboat.co.uk/?p=3132
You may recently have seen people complaining about... some row that went on between Jay, um, Leno and Conan O'Brian on some kind of late-night talk show aired in America. For that matter, you might have seen references to said late night talk shows in general, and be in the situation for which we need a word of "OK, so it's a late night talk show. What's it good at? What's it famous for?"
The linked post does a lovely job of explaining why the row was very, very entertaining, even if you don't normally want to read about dick-waving contests between people you've barely heard of.
"And so began a monumental game of chicken. Conan went onto his show every night subsequently, and did monologues that consisted entirely of calling his network useless, unpopular, incapable of running itself - of calling its executives liars and thieves, and calling Leno all manner of other things"
"it has been, for a brief time, a remarkable insight on the workings of it all - and the true bitterness, fear and anger present in all parts of the industry (of most similar industries) right now."
People with Exactly Two Children and At Least One Daughter, what's the gender make-up of your kids
There's a well-known paradox[1] in statistics, which goes that "suppose you know Mrs Smith has two children, and you see something that suggests that (at least) one of them is a girl. What are the chances that the other one is a boy?"
People instinctively say "50/50"[2]. But in fact, it's 2/3.
This is because there 50% of families have mixed children, 25% two girls, and 25% two boys, so there are twice as many boy/girl families as girl/girl families. You can see this easily if you agree the first child was 50/50 boy/girl, and so boy followed by girl and boy followed by boy are both 25%, as are the other two possibilities.
What's funny is that here: http://boards.straightdope.com/sdmb/showthread.php?t=548550 someone acknowledged that this may not be intuitive to everyone, so made a poll which actually asked people to respond if they had two children, at least one a girl, and say which make-up their family had. It's so much more convincing to see the poll made of actual people, actually showing a 2:1 ratio[3].
PS. Note that this result depends very heavily on finding out by coincidence that one of the children is a girl. A very slight change in wording can make it refer to a situation where the chance IS 50/50, and a very slight change in wording can make it much much more or less obvious which answer is right. In fact, I probably got the wording wrong in my quote, so if you still think it's wrong, it could be that you are right, and my reasoning only applies to the problem I meant to describe.
If you think you've got it understood, next read about the http://en.wikipedia.org/wiki/Monty_Hall_problem with the goats.
Footnote #1: on paradox
Paradox is typically used to mean "an apparently contradictory result". Sometimes this is an ACTUALLY contradictory result, such as "we know time-travel can't allow X because even if we don't know what's physically possible, that's actually self-contradictory". Sometimes this is contradictory because an underlying assumption was subtly false. Sometimes it contradicts what you think OUGHT to be true. In statistics, most paradoxes are of the last sort: all the maths line up, but it superficially looks impossible.
Footnote #2
In these sorts of problems, the aim is to distinguish between two grossly different results stemming from different understandings of the problem. Thus all the actual numbers are approximate: the argument works equally well even though the actual chance of boy/girl is several percent different from 50.
http://littleredboat.co.uk/?p=3132
You may recently have seen people complaining about... some row that went on between Jay, um, Leno and Conan O'Brian on some kind of late-night talk show aired in America. For that matter, you might have seen references to said late night talk shows in general, and be in the situation for which we need a word of "OK, so it's a late night talk show. What's it good at? What's it famous for?"
The linked post does a lovely job of explaining why the row was very, very entertaining, even if you don't normally want to read about dick-waving contests between people you've barely heard of.
"And so began a monumental game of chicken. Conan went onto his show every night subsequently, and did monologues that consisted entirely of calling his network useless, unpopular, incapable of running itself - of calling its executives liars and thieves, and calling Leno all manner of other things"
"it has been, for a brief time, a remarkable insight on the workings of it all - and the true bitterness, fear and anger present in all parts of the industry (of most similar industries) right now."
People with Exactly Two Children and At Least One Daughter, what's the gender make-up of your kids
There's a well-known paradox[1] in statistics, which goes that "suppose you know Mrs Smith has two children, and you see something that suggests that (at least) one of them is a girl. What are the chances that the other one is a boy?"
People instinctively say "50/50"[2]. But in fact, it's 2/3.
This is because there 50% of families have mixed children, 25% two girls, and 25% two boys, so there are twice as many boy/girl families as girl/girl families. You can see this easily if you agree the first child was 50/50 boy/girl, and so boy followed by girl and boy followed by boy are both 25%, as are the other two possibilities.
What's funny is that here: http://boards.straightdope.com/sdmb/showthread.php?t=548550 someone acknowledged that this may not be intuitive to everyone, so made a poll which actually asked people to respond if they had two children, at least one a girl, and say which make-up their family had. It's so much more convincing to see the poll made of actual people, actually showing a 2:1 ratio[3].
PS. Note that this result depends very heavily on finding out by coincidence that one of the children is a girl. A very slight change in wording can make it refer to a situation where the chance IS 50/50, and a very slight change in wording can make it much much more or less obvious which answer is right. In fact, I probably got the wording wrong in my quote, so if you still think it's wrong, it could be that you are right, and my reasoning only applies to the problem I meant to describe.
If you think you've got it understood, next read about the http://en.wikipedia.org/wiki/Monty_Hall_problem with the goats.
Footnote #1: on paradox
Paradox is typically used to mean "an apparently contradictory result". Sometimes this is an ACTUALLY contradictory result, such as "we know time-travel can't allow X because even if we don't know what's physically possible, that's actually self-contradictory". Sometimes this is contradictory because an underlying assumption was subtly false. Sometimes it contradicts what you think OUGHT to be true. In statistics, most paradoxes are of the last sort: all the maths line up, but it superficially looks impossible.
Footnote #2
In these sorts of problems, the aim is to distinguish between two grossly different results stemming from different understandings of the problem. Thus all the actual numbers are approximate: the argument works equally well even though the actual chance of boy/girl is several percent different from 50.
no subject
Date: 2010-01-22 01:50 pm (UTC)A very slight change in wording can make it refer to a situation where the chance IS 50/50
I believe you have it right. If you discover that at least one child of two is a girl there is a 2/3 chance that the other is a boy; however if you discover that the older of two children is a girl then there is a 1/2 chance that the younger is a boy.
[most stats books I have read this in have some stupid 'finding out' thing; relying on crud like "pink stuff is for girls". But supposing you were using rather better information...]
And don't talk to me about goats.
no subject
Date: 2010-01-22 02:48 pm (UTC)Oops. No, that became the PS, but I failed to delete the footnote.
no subject
Date: 2010-01-22 02:53 pm (UTC)I did consider having you see something that would indicate at least one girl. There are plenty of things which are strongly correlated with one gender that aren't offensive. But I decided anything at all was likely to embed some correlation which broke the point of the problem.
Perhaps the most neutral would be if you start asking the question:
"Do you have any girls, Mrs Smith?"
"Oh yes! Hold on, I have to go put the kettle on. With two children, I'm rushed off my feet. What did you say you wanted again?"
Or something.
no subject
Date: 2010-01-22 03:00 pm (UTC)I dunno, it's a bit contrived. But these things always are. Doesn't make the maths any less right.
no subject
Date: 2010-01-22 03:27 pm (UTC)Yeeeeees. But I don't think that would actually work. If you went to a 1000 houses with two children, and glanced casually at a letter from the school whenever you saw one, I think you would probably see "regarding your sons request" between one and two times more often in houses with two sons than with one, which would change the statistics the same was seeing the son, or being told their the eldest would.
no subject
Date: 2010-01-22 09:14 pm (UTC)no subject
Date: 2010-01-22 11:32 pm (UTC)no subject
Date: 2010-01-23 07:52 am (UTC)It's early in the morning, and I've just managed to convince myself both ways in turn.
I still think it's 2/3, but I can't see what makes the difference between my two arguments...
no subject
Date: 2010-01-23 11:36 am (UTC)My reasoning is, consider it two different ways.
1. You know the gender of the child-in-front-of-you. The gender of the other child is 50/50.
2. Consider 100 families of two children. Suppose you meet the mother and one of the children at random from each.
25% of the time you'll meet one boy of two
25% of the time you'll meet one girl of two
12% of the time you'll meet boy from BG
12% of the time you'll meet girl from BG
12% of the time you'll meet boy from GB
12% of the time you'll meet girl from GB
So 50% of the time you'll meet mother and daughter, and half of those will be from GG.
no subject
Date: 2010-01-23 02:04 pm (UTC)I suppose it's that (using your second argument), there are some girl-boy families who you will never know about (because we're never considering the boys who have sisters). Where as in your "accidentally" case, you're assuming that if one of the children is a girl, you always know. Or at least are as likely to know as not know, or something.
no subject
Date: 2010-01-25 02:01 am (UTC)It seems to be whether the thing you see is equally likely to come from one or more girls, or just one. From the first, you'll have representative proportions of BG, GB, and GG. And from the second, you'll have twice as many GG, which brings the odds back to 50/50.
But it can be really hard to tell what's a fair example of "something you might have seen", and it often depends if it's something you decided to go looking for in advance, or something you just happened to see.
no subject
Date: 2010-01-26 10:15 am (UTC)possibly one of the nasty things about that "paradox", then, is that if asked me (or a random passer-by) that question with the phrasing you've given, to solve it I'd come up with a concrete example - such as running into one of the girls - and think of it in a similar way to the second solution you wrote out for me.
[I didn't do that on this occasion, because I know the goat one and have convinced myself in the past, so I skipped through trying to do the maths until I saw the discussion]
And I don't think that the phrasing "see something that suggests that (at least) one of them is a girl" excludes those examples-that-don't-work, so it could be really confusing when the questioner then turns around and says you're wrong! (Ha, I've brought this thread into line with your current post!)
no subject
Date: 2010-01-26 05:42 pm (UTC):)
Yeah, you're right. Even if the question is carefully phrased, that's meaningless unless the recipient is already used to making that sort of distinction.