tag:blogger.com,1999:blog-6894866515532737257.comments2015-08-28T16:19:45.883-07:00Probably Overthinking ItAllen Downeyhttps://plus.google.com/111942648516576371054noreply@blogger.comBlogger467125tag:blogger.com,1999:blog-6894866515532737257.post-35453342721618867262015-08-28T07:42:12.678-07:002015-08-28T07:42:12.678-07:00"The sleeping beauty problem is ambiguous bec..."The sleeping beauty problem is ambiguous because it does not say what sample space she is using." I disagree. Whether or not she will be asked more than once, she only knows about one "questioning" whenever she is asked. And there is only one coin flip. Your alleged ambiguities are red herrings.<br /><br />The problem people encounter is indeed one of sample space, but it is because the sample space OF THE EXPERIMENT has four possible outcomes. They are {MH,MT,UH,UT} where H=heads, T=tails, M=Mondays and U=Tuesdays (I think that is what you wanted to U to be). Tuesday still happens, even if Heads is flipped; SB just does not observe it.<br /><br />To see this better, use four Sleeping Beauties.SB1 will be left asleep under MH, SB2 will be left asleep under MT, SB3 will be left asleep under UH, and will be left asleep under UT. (The original SB is SB3 here.)<br /><br />So three will be wakened on Monday, with the Monday Sleeper swapped for one of them on Tuesday. They will be put into a room together to discuss the question "What is the probability that I will be wakened exactly once?" Each SB knows her own combination, but is prohibited from telling the others.<br /><br />When wakened - as each of them will be - each knows that two of their group will be wakened twice, and one just once, so their answers should sum to 1. Each knows that the information she possess is equivalent to the information possessed by the other two, so their answers have to be equal.<br /><br />Their answers must be 1/3.JeffJohttp://www.blogger.com/profile/09110352332876400907noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-22708623404532384872015-08-27T11:55:37.535-07:002015-08-27T11:55:37.535-07:00Not possible in this case because the detector in ...Not possible in this case because the detector in the basement is wired (not plug in). But your idea is excellent.Allen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-7507316400780844902015-08-27T11:42:50.561-07:002015-08-27T11:42:50.561-07:00Some fine Bayesian reasoning. Experimental desig...Some fine Bayesian reasoning. Experimental design, not so great. Since you have a second detector in the basement, why not immediately tell your wife to go and fetch it? Two independent detectors going off upstairs (but not in the basement) updates your probabilities to close to 1 and 0 immediately.Ralph Burgesshttp://www.blogger.com/profile/15987132500150157640noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-28295968206917744422015-08-27T07:05:04.478-07:002015-08-27T07:05:04.478-07:00She already knew on Sunday that she would not be i...She already knew on Sunday that she would not be interviewed on Tuesday if the coin toss was heads. She didn't learn anything new on waking.Brian Mayshttp://www.blogger.com/profile/13962229896535398120noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-20449532603078764592015-08-24T09:48:48.945-07:002015-08-24T09:48:48.945-07:00> Once you notice the inspection paradox, you s...> Once you notice the inspection paradox, you see it everywhere. Does it seem like you can never get a taxi when you need one? <br /><br />And of course, besides buses, taxis, and planes, there's cars: why does the highway so often seem to be jammed? Because if it's jammed, there's a lot more people in it than when it's moving smoothly; I saw this version in Nick Bostrom's http://www.anthropic-principle.com/?q=book/table_of_contents<br /><br />Which makes me wonder if there is any mode of transportation that the inspection paradox *doesn't* apply to... perhaps when the bucket or lump size equals one and so there is no difference in sampling, like walking or riding your bike or roller skating?gwernhttp://www.blogger.com/profile/18349479103216755952noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-29045631900805812002015-08-23T14:01:56.817-07:002015-08-23T14:01:56.817-07:00Yes, PDFs are definitely easier to read for most p...Yes, PDFs are definitely easier to read for most people. The problem is that if you have discrete data, you have to smooth the PDFs to see the shape, and if you smooth too much, you obliterate the bimodality. And even if you get it right, it is harder to compare multiple PDFs to see the kind of shifts I want to show.<br /><br />But since several people have asked, I might try again to make PDF-based visualizations for these examples and see if I can make them work.Allen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-55653221147342454612015-08-23T13:58:26.521-07:002015-08-23T13:58:26.521-07:00Good point. I will revise this and talk about cha...Good point. I will revise this and talk about changes of demand rather than supply, which I think will be clearer and avoid the issue you raised.Allen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-89290212114456054602015-08-23T13:57:07.581-07:002015-08-23T13:57:07.581-07:00Good one. Thanks!Good one. Thanks!Allen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-29333916686423653102015-08-22T18:36:34.396-07:002015-08-22T18:36:34.396-07:00Very nice indeed. I shared this via G+ and got a l...Very nice indeed. I shared this via G+ and got a lot of very positive reader feedback on your article as well. (https://plus.google.com/+YonatanZunger/posts/FjCWr1SySQg)<br /><br />One question: Does it make sense to use PDF's instead of CDF's to illustrate this? With the jogging example (figure 4) in particular, I think a PDF would have highlighted the bimodality far more vividly. (As well as being much more accessible to a non-specialist audience)Yonatan Zungerhttp://www.blogger.com/profile/08967761812729609699noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-64154008758422202322015-08-22T15:03:50.395-07:002015-08-22T15:03:50.395-07:00Worth reading. I recall when the PC and Windows f...Worth reading. I recall when the PC and Windows first hit the business desktop, being dumbfound by users tolerance of the rate at which they crashed. <br /><br />At the time I was accustomed to mature IBM AS/400 super mini hardware that had an MTBF of about 5 years, and I realized that when PC crashed, it was just one person who was impacted by it, but when the main computer for the whole company crashed, everyone experienced the pain at the same time. Even though the total man-hours a PC was down was orders of magnitude greater than the man-hours the AS/400 was down, the distribution of the pain a little at a time made is seem like less. Roger Weberhttp://www.blogger.com/profile/13039129656717602178noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-52294638681477720082015-08-21T10:19:48.004-07:002015-08-21T10:19:48.004-07:00In Hong Kong the Chinese believe that boys come ea...In Hong Kong the Chinese believe that boys come earlier than girls. For boys they come within 36-38 weeks and for girls between 37-40 weeks. I no actual data, it may be interesting to check this out.fengshui leunghttp://www.blogger.com/profile/06143376466358780225noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-19611299263780419272015-08-21T10:02:29.806-07:002015-08-21T10:02:29.806-07:00nice piece, thx.nice piece, thx.fengshui leunghttp://www.blogger.com/profile/06143376466358780225noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-34782492549218460422015-08-21T05:27:14.784-07:002015-08-21T05:27:14.784-07:00If [there] are 10 students in a class, you have 10...If [there] are 10 students in a class, you have 10 chances to sample that class.<br /><br />- great article!Jimhttp://www.blogger.com/profile/15958070809506087817noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-5857198757792868972015-08-21T04:15:51.682-07:002015-08-21T04:15:51.682-07:00Here's something you might want to clarify:
&...Here's something you might want to clarify:<br /><br />"when there is a surplus of taxis, only a few customers enjoy it. When there is a shortage, many people feel the pain." <br /><br />This is true if the number of taxis stays constant and the people vary; but if the number of people stays constant and the taxis vary, it doesn't hold.4verageJo3http://www.blogger.com/profile/08012486411258768407noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-34215916962349410752015-08-21T00:41:27.297-07:002015-08-21T00:41:27.297-07:00Even after having read the article, it took me 10 ...Even after having read the article, it took me 10 minutes to figure it out ;)Pierre Mhttp://www.blogger.com/profile/09604463877214596048noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-19634454598735605352015-08-20T19:30:11.512-07:002015-08-20T19:30:11.512-07:00Allen, nice article...well-written and easy to und...Allen, nice article...well-written and easy to understand. One suggestion: use "interarrival time" in the train example instead of "arrival time" because the former is standard queueing parlance. Thanks. craig.froehlehttp://www.blogger.com/profile/08965300451627112100noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-1514687840203474192015-08-20T12:29:23.212-07:002015-08-20T12:29:23.212-07:00Nice post. An example from my own field: The inspe...Nice post. An example from my own field: The inspection paradox causes most students to get the wrong answer by a factor of 2 in a certain physics calculation, related to electron motion in solids - see http://physics.stackexchange.com/questions/88015/definition-of-mean-free-time-in-the-drude-model/88045#88045Stevehttp://www.blogger.com/profile/09879280683468964072noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-50092389127385245292015-08-20T02:17:32.081-07:002015-08-20T02:17:32.081-07:00This is a fascinating post, thanks.This is a fascinating post, thanks.Nunohttp://www.blogger.com/profile/09089984219014317886noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-49445362627735239232015-08-19T12:51:15.188-07:002015-08-19T12:51:15.188-07:00Good point. Similarly with the relay race example...Good point. Similarly with the relay race example, you can't tell how many runners are the same speed as you.Allen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-73975504488638017212015-08-19T12:37:01.555-07:002015-08-19T12:37:01.555-07:00yeah, the expected class size of a random student ...yeah, the expected class size of a random student is not the sum of the class sizes divided by the number of classes, but rather the sum of the squared class sizes divided by the sum of the class sizes.<br /><br />good examples though. another very obvious one is the speed of cars you pass on the highway.<br /><br />be aware of one limitation to your inversion: by looking at the samples, you cannot determine the number of classes with zero students, so you'll never know the actual average class size.Roel van Heeswijkhttp://www.blogger.com/profile/08198557649340374292noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-80526792786786166342015-08-19T00:42:51.294-07:002015-08-19T00:42:51.294-07:00My favorite example (true in many European countri...My favorite example (true in many European countries): most families have a single child, but most kids have siblings.luispedrohttp://www.blogger.com/profile/05396839866731499408noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-69561922660365901242015-08-18T13:57:21.577-07:002015-08-18T13:57:21.577-07:00I totally understand, and I am sympathetic -- I th...I totally understand, and I am sympathetic -- I think most people are more familiar with PDFs than CDFs. But for these examples I have empirical data rather than continuous mathematical models, and in that case using PMFs or estimated PDFs creates a whole bunch of problems that CDFs neatly avoid.<br /><br />I decided it was better to use CDFs throughout. But it means the reader has to work a little harder, which I regret.Allen Downeyhttp://www.blogger.com/profile/01633071333405221858noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-74955440819612802992015-08-18T12:35:42.541-07:002015-08-18T12:35:42.541-07:00Thanks a lot, very informative, and enjoyable. A m...Thanks a lot, very informative, and enjoyable. A minor quibble: I find it easier to comprehend densities rather than cumulative distribution functions. Perhaps I can sway you to illustrate distributions by their pdfs instead of cdfs.Robert Dodierhttp://www.blogger.com/profile/09299746495471300195noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-18773026705963699632015-08-16T16:29:59.735-07:002015-08-16T16:29:59.735-07:00The sleeping beauty problem is ambiguous because i...The sleeping beauty problem is ambiguous because it does not say what sample space she is using. Probabilities are defined on a per sample space basis. The sample space of the coin toss is {H,T} and the sample space for the questions about the coin state is {MH,MT,UT} where H=heads, T=tails, M=Mondays and T=Tuesdays. The probability of heads for the first sample space is 1/2 and the probability of heads for the second sample space is 1/3, since they are both equiprobable sample spaces. To see equiprobability, just notice that out of every 1000 coin tosses about 500 will be heads, 500 will be tails, and about 1500 questions will be asked about 500 which will occur when it is Monday and heads, another 500 which will occur when it is Monday and tails, and the remaining 500 which will occur when it is Tuesday and tails.<br /><br />She should use the probability for the sample space she assumed and the problem doesn't tell what sample space that is. The problem is bad because it introduces two different sample spaces without clarifying which one is operative. For example, if the problem also stated that for betting purposes on repeated trials of the experiment she should bet as much money as possible then it would be clear that she should use the sample space for the questions about the coin state to get the probability. But if instead of that we added to the original problem that she give the probability for repeated tosses of the coin then money won or lost is irrelevant and she should use the sample space for the coin toss to get the probability. The sleeping beauty question is ambiguous because it is asking about belief in the frequency of the truth value of occurrences of the PROPOSITOIN "the coin landed heads" not the proposition that the coin's probability of landing heads is 1/2. That is, the question doesn't make clear if it is asking about the probability of the proposition being true during repeated coin tosses or if it is asking about the probability of the proposition being true during repeated questioning in many repetitions of the experiment. These are not the same thing because when the coin is tails she is questioned twice but when the coin is heads she is questioned only once.<br /><br />Now she knows the proposition is true one out of every three times she is asked and she is not going to mistake that for the fact that the coin comes up heads one out of every two times during coin tossing. So, for the proposition "the coin landed heads" the frequency of this proposition being true during repeated questioning in many repetitions of the experiment is different than the frequency of it being true during repeated coin tosses. If the coin toss actually came up heads then the proposition "the coin landed heads" is true but if the coin toss actually came up tails then the proposition "the coin landed heads" is false. How often the proposition is true or not depends on the circumstances. So adding two different prepositional phrases onto the original question highlights the ambiguity of that question:<br /><br />(case 1)<br />What is your belief now for the proposition that "the coin landed heads" in the case of repeated questioning in repetitions of the experiment?<br /><br />(case 2)<br />What is your belief now for the proposition that "the coin landed heads" in the case of repeated tosses of the coin?<br /><br />The conclusion: the sleeping beauty problem is ambiguous because case 1 and case 2 use different sample spaces and if one removes the phrase "questioning in repetitions of the experiment" from case 1 and removes the phrase "tosses of the coin" from case 2 then the ambiguity of the original question is exposed.Louis Wilburhttp://www.blogger.com/profile/17153145669881153091noreply@blogger.comtag:blogger.com,1999:blog-6894866515532737257.post-91309522653825224362015-08-04T11:51:47.228-07:002015-08-04T11:51:47.228-07:00Nora: No, the child has a 50% chance to be a girl....Nora: No, the child has a 50% chance to be a girl. You have trouble understanding Mlodinow, because he answers different questions than what he asks. Those questions are:<br /><br />"If you pick one family at random from the set of all two-child families that include a girl, what are the chances that family has two girls?"<br /><br />"If you pick one family at random from the set of all two-child families that include a girl named Florida, what are the chances that family has two girls?"<br /><br />You noticed an increase in the probability the "other" child is a girl if the "first" is named Florida. That is caused by the fact a two-girl family is twice as likely to have one named Florida, than a one-girl-one-boy family would be. So while there are twice as many one-girl-one-boy families as two-girl families in the set for the first question, the second's set has about (see note below) the same number of each.<br /><br />The answers to the questions he asked are both 1/2. Not 1/3 and almost 1/2. That's because, if you merely remember a fact about one child from a one-girl-one-boy family, there is a 50% chance it will be about the boy. So the sets Mlodinow should have used would include only the half of them where you remember the girl.<br /><br />Note: There are actually more two-girl families in the set. Mlodinow got that there were less because he not only allowed families to have two Florida's, but also two Mary's, two Sue's, two Ann's, etc. When you don't do that, the chances that a second girl will have any particular name go down a little for common names, and go up a little for uncommon ones. But "little" is relative - it can multiply the chances for very uncommon names, like Florida, by hundreds or thousands.JeffJohttp://www.blogger.com/profile/09110352332876400907noreply@blogger.com