tag:blogger.com,1999:blog-3500036.post7893380644507741502..comments2020-02-24T00:22:55.494-05:00Comments on He Lives: Tuesday Child PuzzleDavidhttp://www.blogger.com/profile/08688240424047203541noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-3500036.post-63186237873679166482010-08-26T16:21:40.743-04:002010-08-26T16:21:40.743-04:00I remember John Derbyshire demonstrating this same...I remember John Derbyshire demonstrating this same problem on the Corner not long ago. Although I didn't bother to work it out, I *did* figure out why the answer wasn't intuitive (namely, that people confuse the question with "What are the chances that my next child will be a boy?"). Weirdly, he didn't seem to, or at least he didn't let on that he had, even though he showed the right answer.The Deucenoreply@blogger.comtag:blogger.com,1999:blog-3500036.post-33861287526569418832010-08-22T21:05:34.263-04:002010-08-22T21:05:34.263-04:00Nice explanation at 8:01AM. The ambiguity of trans...Nice explanation at 8:01AM. The ambiguity of translating English to mathematics leads to many of these probability "paradoxes." Even when the original story seems straightforward, as is this case, the conditional probabilities, which depend on unstated assumptions, are not always precisely defined by the story. E.g. cases #1 and #2.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3500036.post-51586023889625560382010-07-27T08:01:13.958-04:002010-07-27T08:01:13.958-04:00Unfortunately, these questions you ask are ambiguo...Unfortunately, these questions you ask are ambiguous, and it is the failure to recognize how they are ambiguous that causes the results to seem unexpected. Consider two versions of what led up to the first statement:<br /><br />Case #1: A father is chosen at random. He is given a slip of paper as he is led onto a stage. The paper says "Pick one of your children. Tell the audience the number of children you have, the chosen child's gender, and the day of the week on which it was born."<br /><br />Case #2: A father is chosen at random from all fathers who have two children, including one boy born on a Tuesday. He is also ushered onto a stage and given a slip of paper that instructs him to tell the audience the criteria used to select him.<br /><br />Now shift scenes. You are in the audience when a man is ushered onto the stage. He looks at a slip of paper, thinks a moment, and says "I have two children and one of them is a boy born on Tuesday." What is the probability that he has two boys?<br /><br />The answer to the question depends on which case applies to the man you listened to. In Case #1, it is 1/2. In case #2, it is 13/27. Your simulation only covered the second case. To get the first, after you have two children, flip a coin to see which one the father will tell about. If it is not a Tuesday Boy, don’t keep that trial even if the other child is a Tuesday Boy. You will find that the 27 cases where you have a Tuesday Boy reduce to 14 (just over half, since one father didn’t need to flip the coin), the 13 where you also have two boys reduces to 7, and the answer is exactly 1/2.<br /><br />If you simulate the simpler problem, where you don’t worry about the day of the week, the answers are 1/2 and 1/3 for the two cases, respectively. The reason 13/27 seems unintuitive, is because the fact that a Tuesday Boy was REQUIRED in the second case is not intuitively obvious from the statement "one of them is a boy born on Tuesday." In fact, as you point out, the puzzle could equally well be named after either of your two children, which is probably two different names. You choose one, just like the father in case #1, so the better answer to your question is 1/2, not 13/27. It is still ambiguous, but there is no valid reason to assume that case #2 applies.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3500036.post-23705374798593685402010-07-25T23:58:06.925-04:002010-07-25T23:58:06.925-04:00This is an intriguing puzzle, prompting me to offe...This is an intriguing puzzle, prompting me to offer an excel-based precise solution along with delivery tips at <a href="http://timwit.wordpress.com/2010/07/25/tuesday-child-puzzle-and-superiority-trick/" rel="nofollow">this site</a>timwitnoreply@blogger.comtag:blogger.com,1999:blog-3500036.post-55016876142952244212010-07-23T17:39:43.799-04:002010-07-23T17:39:43.799-04:00My restaurant has 14 items on the menu: Burger, bu...My restaurant has 14 items on the menu: Burger, burger w/cheese, chicken, chicken w/cheese, fish, fish w/cheese, pulled pork, pulled pork w/cheese, tofu, tofu w/cheese, alligator, alligator w/cheese, goat, and goat w/cheese. <br /><br />Three questions:<br /><br />I have an order in for two items. One of the items has cheese. What are the odds the other item has cheese? Answer = 1/3<br /><br />I have an order in for two items. One of the items is a burger w/ cheese. What are the odds the other item has cheese? Answer = 13/27<br /><br />I have an order in for two items. One of the items has cheese. What are the odds the other item has cheese? Oh by the way, the aforementioned item is a burger. Answer = STILL 13/27.<br /><br />Right?James Sweethttps://www.blogger.com/profile/17212877636980569324noreply@blogger.comtag:blogger.com,1999:blog-3500036.post-55683550308074187862010-07-23T17:35:49.865-04:002010-07-23T17:35:49.865-04:00Okay, Bob O'H, I've got a rebuttal.
I am ...Okay, Bob O'H, I've got a rebuttal.<br /><br />I am going to randomly pick letters out of a hat, where the possibilities are "A", "a", "B", and "b". All are equally probably. One of the letters is a lower-case "a". What is the probability that the other letter is an "A/a", i.e. either case?<br /><br />P(aa) = 1/16<br />P(Aa) = 1/16<br />P(aA) = 1/16<br />P(ba) = 1/16<br />P(ab) = 1/16<br />P(Ba) = 1/16<br />P(aB) = 1/16<br />------------<br />Total = 7/16<br /><br />P(aa)+P(Aa)+P(aA) = 3/16<br /><br />So the answer to the question is 3/7. With me so far?<br /><br />Now, would your answer change if I phrase the question like this:<br /><br />I am going to randomly pick letters out of a hat, where the possibilities are "A", "a", "B", and "b". All are equally probably. One of the letters is some type of A. What is the probability that the other letter is an "A/a", i.e. either case? Oh by the way, the first letter I mentioned is lower-case.James Sweethttps://www.blogger.com/profile/17212877636980569324noreply@blogger.comtag:blogger.com,1999:blog-3500036.post-46732122829166551822010-07-23T17:06:28.563-04:002010-07-23T17:06:28.563-04:00I'm just figuring this out myself... but the a...I'm just figuring this out myself... but the apparently paradoxical answer does not require that it eliminates the possibility that there is a second boy born on Tuesday.<br /><br />The point is that if the other child is not a boy born on Tuesday, then it could have been either the first or the second child. So the probability of those outcomes is effectively "doubled" (I'm speaking loosely here). But if the other child is a son born on Tuesday, then we know for a fact that the first child had to be a son born on Tuesday -- which means we don't get to do the double-counting I alluded to earlier.<br /><br />It really bothers me, but I think this is correct.James Sweethttps://www.blogger.com/profile/17212877636980569324noreply@blogger.comtag:blogger.com,1999:blog-3500036.post-10212417099169986152010-07-21T09:17:48.801-04:002010-07-21T09:17:48.801-04:00This comment has been removed by a blog administrator.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3500036.post-18573523481730365072010-07-20T11:11:05.847-04:002010-07-20T11:11:05.847-04:00Bob O'H said it: unless saying "One is a...Bob O'H said it: unless saying "One is a boy born on a Tuesday" eliminates the possibility that there is a second boy, also born on a Tuesday, which it doesn't seem to logically, then the answer would remain 2/3.zilchhttps://www.blogger.com/profile/01695741977946935771noreply@blogger.comtag:blogger.com,1999:blog-3500036.post-50980627824259257452010-07-11T13:23:43.249-04:002010-07-11T13:23:43.249-04:00Hm. I'm not convinced: the boy has to be born ...Hm. I'm not convinced: the boy has to be born on some day, so that day can be reported.<br /><br />The problem is really semantics, not maths. The question can be read as saying "I have two children. One is a boy. What is the probability I have two boys? BTW, the boy referred to was born on Tuesday". That would imply that Tuesday is irrelevant. The question is poorly worded, has 3 or 4 'correct' answers, depending on how it is interpreted.<br /><br />FWIW, I think 2/3 is the right answer.Bob O'Hhttps://www.blogger.com/profile/15666738696003108444noreply@blogger.comtag:blogger.com,1999:blog-3500036.post-29428224763835530892010-07-10T20:35:52.066-04:002010-07-10T20:35:52.066-04:00This comment has been removed by a blog administrator.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3500036.post-35033877563377341392010-07-08T21:07:57.913-04:002010-07-08T21:07:57.913-04:00Wow, this is pretty neat. Even when you see it la...Wow, this is pretty neat. Even when you see it laid out, it still isn't intuitive. I had a statics professor bring up the "Let's make a deal" problem a few semester ago. I still can't convince anyone that swapping doors is the best choice.Lardernoreply@blogger.com