tag:blogger.com,1999:blog-3317748811909502538.post6364904388077997405..comments2024-02-19T09:39:33.277+00:00Comments on Intropy: Tuesday's child (or why New Scientist has got it wrong)DaveoftheNewCityhttp://www.blogger.com/profile/04140446220455064332noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3317748811909502538.post-11206292415269124382013-03-18T21:13:05.880+00:002013-03-18T21:13:05.880+00:00PS, I copied Vincent's comment into a later b...PS, I copied Vincent's comment into <a href="http://www.intropy.co.uk/2013/01/probability-and-sexism.html" rel="nofollow">a later blog post</a>David Chapmanhttps://www.blogger.com/profile/01959069828311977846noreply@blogger.comtag:blogger.com,1999:blog-3317748811909502538.post-69952088070611365582013-03-18T21:09:46.704+00:002013-03-18T21:09:46.704+00:00Thank you for your comment, Jeff, it is fascinatin...Thank you for your comment, Jeff, it is fascinating how much there is to this. <br /><br />In a similar case (though with Knights hunting dragons!) Chris Maslanka in his puzzle column in the Guardian argued a couple of weeks back that you could not give an answer for the probability, because you don't have enough information. You don't know what he's meaning when he says he has one male dragon. <br /><br />However, I've now thrown the newspaper away and I don't think the puzzles appear in the online version of the paper so I can't check the details.David Chapmanhttps://www.blogger.com/profile/01959069828311977846noreply@blogger.comtag:blogger.com,1999:blog-3317748811909502538.post-18253014260560025932013-03-12T21:15:36.877+00:002013-03-12T21:15:36.877+00:00. This apparently got "bumped" on the se.... This apparently got "bumped" on the search engines because of Vincent's comment, and I just found it. David Chapman had the right idea, but didn’t carry it far enough. To illustrate, I'm going to change the problem in a couple of ways that have absolutely no impact on how we should get to the answer. But I also need to point out that Gary Foshee asked this question at a conference honoring Martin Gardner, the long-time author of Scientific American's "Mathematical Games" column.<br /><br />You run across an old school buddy - you were both in the Mathematical Games Club - on the street. While getting reacquainted, he tells you "I have two children, and one of them is a XXXXX. Given that information about one gender, what is the probability I have a boy and a girl?" But in the middle of his statement, a passing driver honked his horn, drowning out the gender your friend named (as represented by XXXXX). Can you still answer?<br /><br />If you had actually heard your friend say "boy," then Gary Foshee says the answer is 2/3 (one minus the chance of two boys). Gary Foshee says this, because he remembers that Martin Gardner once said it for a similar problem. And I think, but it isn't clear, that David Chapman would agree.<br /><br />But now we have a paradox. Because you would also have to say 2/3 if the obliterated word was "girl." And if the answer is 2/3 regardless of which word was obliterated, it has to be 2/3 even if your friend hadn't made the statement. Yet we know that probability is 1/2.<br /><br />This is a variation of a famous problem introduced by Joseph Bertrand in 1889, called Bertrand's Box Paradox. His point, much like the one David didn’t take far enough, was that knowledge of the possible combinations is not enough to solve a problem like this. You also need to know why this particular fact was given to you, when other equivalent facts could have been.<br /><br />If your friend actually has two boys, or two girls, there were no alternatives. But if he has one of each, he had to choose. And if you don't know how he made that decision, you can only assume he choose randomly. That is, he had a 50% chance to say "boy," and a 50% chance to say "girl." And that means that the answer Foshee and David should have is 1/2, not 2/3. Yes, 2/3 of all two-child families at least one with a boy have two, but only half of the fathers who would randomly *tell* you about a boy have two. Ironically, Foshee seems to have missed that Martin Gardner changed his answer in that recalled column for this reason.<br /><br />And it doesn't matter if other facts are included in the XXXX that you didn't hear. The answer is always 1/2 - exactly, given the assumptions about children's genders - no matter what included. Vincent was being generous is claiming that 1/3 (for the original question) was equally correct as 1/2. 1/3 can only be correct if, as David describes for birth-days but didn’t take far enough, the selected father was forced to tell you about a boy if he could.<br />JeffJohttps://www.blogger.com/profile/09110352332876400907noreply@blogger.comtag:blogger.com,1999:blog-3317748811909502538.post-63515348647779803652013-01-20T00:04:54.475+00:002013-01-20T00:04:54.475+00:00I know it's been two years since this post, bu...I know it's been two years since this post, but I came back to it recently after looking at one of your other posts and it got me thinking again.<br /><br />After a few minutes of trying to place my finger on exactly why the statement that one of the children is born on Tuesday seems to affect the probability of the other child being a boy, I realized that the exact same logic can be applied to the original "two boy puzzle". <br /><br />The amazing truth is, that those people who automatically say the answer to the two boy puzzle is 1/2, are actually equally as correct as those who answer 1/3. The thing is, that somebody who walks up to you and says "I have two children. One of them is a boy. What is the probability I have two boys?" could have said "I have two children. One of them is a girl..." if they have a girl, but they would be forced to say they had a boy if they had two boys. Hence, if we assume that the person we are talking to is not sexist, then the probability that they have two boys, given that they volunteered the fact that they have a boy is actually twice the chance of each of the two instances of them having a girl. Hence the probability that the other child is a boy is equal to the probability that the other child is a girl and hence 1/2.<br /><br />The pitfall of the majority of statisticians who claim the answer to this puzzle is 1/3, is that they have taken each possible permutation of the genders of the two children to be equally likely events, something which we simply cannot presume to be the case. Amusingly the actually answer ranges depending on how sexist the people around you are, where 1/3 is the answer in the case where the person would only ever volunteer the fact they have a girl if they didn't have any boys to brag about and 1/2 is the answer in the case where the person would randomly volunteer boy or girl if they had a boy and a girl with equal probability. The final case is where the person is entirely biased against boys, and in this case we could be absolutely sure that the other child is a boy, otherwise they would have told us they have a girl.<br /><br />This conclusion links in quite nicely to ideas of information. A sexist person conveys more information about the gender of the other child by the choice they made of which child to volunteer, in the case where they have a boy and a girl. The only case where no information is conveyed is when the child is picked randomly. This is a similar conclusion to the one we came to when talking about the Game-Show Earthquake problem.<br /><br />No doubt some people would shoot me down when I say this but I'm absolutely convinced that the actual answer to the two boy puzzle is in fact 1/2, despite it seemingly being taken as fact that the answer is 1/3. I say this because in REALITY, if someone walked up to you and asked this question the probability would be close to 1/2. Although ironically now that this puzzle has circulated, anyone who asked this question will probably be more likely to say boy because they know it as the "two boy puzzle" rather than the "two girl puzzle".<br /><br />This also demonstrates what I always say about people misunderstanding statistics, even people who are absolutely sure they understand it, because those who answer 1/3 to the puzzle have made a similar mistake to those people who say things like "wow, if only I'd bet on that happening before the match! I'd be a millionaire!" The answer being well yes you would, but you'd have to have made presumptions before the match about something you simply couldn't have known was going to happen. In a similar fashion, we've made presumptions about the nature in which the question has been asked when we simply can't do so.Anonymousnoreply@blogger.com