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0|0|0|||||Bear born on Tuesday problem|Julian Franklin|1|10:48:37|03/31/2011|The problem was originally stated as%3A%0D%0A%0D%0A%22If%2C say%2C it is known that one of the bears is a male born on a Tuesday%2C what is the probability of the other bear to be also male%3F%22%0D%0A%0D%0AThat seems to establish the bear born on Tuesday as an independant event %28i.e. established PRIOR to our work determining the probability of the other bear%27s sex%29. But the math that follows seems to determine the probability of BOTH events %28i.e. the odds of two male bears at least one of which was born on Tuesday%29%0D%0A%0D%0AIf I say %22I flipped a coin five times and it came up heads%2C what are the odds of the next coin flip coming up heads%3F%22%2C the odds are 50%2F50. In the coin problem the previous flips are independant events and have no bearing on the outcome of the sixth toss. How is the bear problem %28the way it is worded%29 different%3F%0D%0A
2|1|0|||||RE%3A Bear born on Tuesday problem|alexb||10:54:04|03/31/2011|What parallels the problem in terms of coins is this%3A%0D%0AA coin is thrown twice. Heads show up on the first throw.%0D%0A%0D%0AWhat is the probability that Heads will show up%0D%0A%0D%0A%5Bol%5D%0D%0A%5Bli%5Din the second throw%2C%0D%0A%5Bli%5Din two throws.%0D%0A%5B%2Fol%5D%0D%0A%0D%0AThese two are equal for the very reason that it is known that on the first through heads came up.%0D%0A%0D%0AThe probability of heads in the second throw is of course 1%2F2 independent of the reault of the first throw.
3|2|2|||||RE%3A Bear born on Tuesday problem|Julian Franklin|1|22:05:35|04/02/2011|Yes%2C I agree. The first coin tosses have no bearing on the future coin tosses. They are independant events because the results of the first coin toss was established.%0D%0A%0D%0ASo%2C when the problem is presented as %22I have two bears%2C one is a male born on Tuesday%2C what are the odds the second bear is a male%22 then that seems to establish %22one male born on Tuesday%22 as a %22given%22%2C just like the first coin toss in our example.%0D%0A%0D%0ANow all we have to do is determine the probability of the second bear being male%2C not of finding two males at least one of which was born on Tuesday.%0D%0A%0D%0AIt would seem that if we wanted to find the second probability%2C then the problem should be stated differently%3A%0D%0A%0D%0A1%29 %22We have two bears. What are the odds that both of them are male and at least one was born on Tuesday%3F%22%29%0D%0A%0D%0Aor%0D%0A%0D%0A2%29 %22We have two bears. One is male. What are the odds that the other is also male and that at least one of them was born on a Tuesday%3F%22.%0D%0A%0D%0AI recognize that both of those questions result in different answers%2C both different from the answer to%0D%0A%0D%0A3%29 %22We have a two bears%2C one is a male born on Tuesday%2C what are the odds that the other bear is also a male%3F%22 %28which is what I understand the question on the forum board to ask%29%0D%0A%0D%0ASo I guess I%27m wondering which of these three questions is really being posed because it seems like it is the third%2C but the answer more accurately seems to reflect the second.
4|3|3|||||RE%3A Bear born on Tuesday problem|alexb||22:37:08|04/03/2011|%3EYes%2C I agree. The first coin tosses have no bearing on the %0D%0A%3Efuture coin tosses. They are independant events because the %0D%0A%3Eresults of the first coin toss was established. %0D%0A%0D%0AWhy%2C they are independent regardless of whether the first tos was established or not.%0D%0A%0D%0A%3ESo%2C when the problem is presented as %22I have two bears%2C one %0D%0A%3Eis a male born on Tuesday%2C what are the odds the second bear %0D%0A%3Eis a male%22 then that seems to establish %22one male born on %0D%0A%3ETuesday%22 as a %22given%22%2C just like the first coin toss in our %0D%0A%3Eexample. %0D%0A%0D%0ANo%2C we do not know which bear - first or second - is a male born on a Tiesday. %22One is a male%22 is not the same as %22First is a male.%22 Perhaps%2C I should have asked about the other  - not the second - bear.%0D%0A%0D%0A%3ENow all we have to do is determine the probability of the %0D%0A%3Esecond bear being male%2C not of finding two males at least %0D%0A%3Eone of which was born on Tuesday.%0D%0A%0D%0AAgain%2C we do not know to which bear the condition applies.%0D%0A%3E%0D%0A%3EIt would seem that if we wanted to find the second %0D%0A%3Eprobability%2C then the problem should be stated differently%3A %0D%0A%3E%0D%0A%3E1%29 %22We have two bears. What are the odds that both of them %0D%0A%3Eare male and at least one was born on Tuesday%3F%22%29 %0D%0A%0D%0AThis is certainly a different problem.%0D%0A%0D%0A%3Eor %0D%0A%3E%0D%0A%3E2%29 %22We have two bears. One is male. What are the odds that %0D%0A%3Ethe other is also male and that at least one of them was %0D%0A%3Eborn on a Tuesday%3F%22. %0D%0A%0D%0AThis is also a different problem.%0D%0A%0D%0A%3EI recognize that both of those questions result in different %0D%0A%3Eanswers%2C both different from the answer to %0D%0A%3E%0D%0A%3E3%29 %22We have a two bears%2C one is a male born on Tuesday%2C what %0D%0A%3Eare the odds that the other bear is also a male%3F%22 %28which is %0D%0A%3Ewhat I understand the question on the forum board to ask%29 %0D%0A%0D%0ARight. I should be using the word %22other%22 rather than the word %22second.%22%0D%0A%0D%0A%3ESo I guess I%27m wondering which of these three questions is %0D%0A%3Ereally being posed because it seems like it is the third%2C %0D%0A%3Ebut the answer more accurately seems to reflect the second. %0D%0A%0D%0AI may have applied a different reasoning. After counting the number of fm pairs that is 14%2C assume first that the mail bear born on a Tuesday is the first one %28M%29. Then we have 7 pairs Mm. In the second case%2C we also have 7 pairs mM. But observe that the pair MM is included in both Mm and mM. So that we have only 13 %3D 7 %2B 7 - 1 male pairs. %0D%0A%0D%0AThe table simply helped save a few sentences.
5|4|4|||||RE%3A Bear born on Tuesday problem|alexb||22:40:47|04/03/2011|Just checked%2C I did ask about the %22other%22.
6|5|5|||||RE%3A Bear born on Tuesday problem|Julian Franklin|1|11:24:05|04/05/2011|First off%2C I appreciate your time replying to me and I really enjoy these sorts of issues%2C so thank you for the site and your time to engage back and forth on these questions.%0D%0A%0D%0AI still don%27t see how it makes a difference which bear is the %22one%22 and which is %22the other%22 %28though I do agree you said %22the other%22 and not %22the second%22 and I realize little things like this often do make a difference so I am sorry for misquoting%29.%0D%0A%0D%0ALet%27s take this premise as an example%3A %22I have two coins%2C one of them is a coin that I tossed yesterday and it came up heads%2C what are the odd that the other coin is also heads%22.%0D%0A%0D%0AWouldn%27t it be safe to assume that it doesn%27t matter WHEN the established coin was tossed%3F Isn%27t it clearly an independant event whether it was tossed yesterday%2C last Tuesday%2C five years ago or five seconds ago%3F %0D%0A%0D%0AConsider that if the %22Male Bear on Tuesday%22 problem is true as stated on the website%2C then the probability of %22the other bear%22 being male changes depending on how we measure the birthdate of the bear we know to be male. Consider these variations on this problem%3A%0D%0A%0D%0A1%29 Two bears%2C one is a male born on Tuesday. What are the odds the other bear is male%3F%0D%0A   13%2F27 --%3E  48.15%25 %28rounded%29%0D%0A%0D%0A2%29 Two bears%2C one is a male born in August. What are the odds the other bear is male%3F%0D%0A   144%2F564 --%3E  25.53%25 %28rounded%29%0D%0A%0D%0A3%29 Two bears%2C one is a male born on August 22. What are the odds the other bear is male%3F%0D%0A   133%2C225%2F532%2C535 --%3E 25.02%25 %28rounded%29%0D%0A%0D%0A4%29 Two bears%2C one is a male born on an even numbered day of the month. What are the odds the other bear is male%3F%0D%0A   12%2F20 %3D 3%2F5 --%3E 60%25%0D%0A%0D%0A5%29 Two bears %28from another social structure where weeks all have ten days each%29 and one bear is a male born on Two-sday. What are the odds the other bear is male%3F%0D%0A   100%2F390 %3D 10%2F39 --%3E 25.64%25 %28rounded%29%0D%0A%0D%0A%28FYI%3A For these calculations I pretended that the days were evenly distributed among the 12 months and among the even vs. odd days in order to simplify the calculations%29.%0D%0A
7|6|6|||||RE%3A Bear born on Tuesday problem|alexb||11:36:37|04/06/2011|I absolutely agree with you%2C except that I do not see a contradiction in that the probability changes depending on the kind of information that is known at the time the probabilities are calculated. Specifics modify the sample space which leads to different probability estimates.%0D%0A%0D%0AThe coin analogy does not spans out. What happened - happened. No question about that. 
8|7|7|||||RE%3A Bear born on Tuesday problem|Julian Franklin|1|11:55:36|04/09/2011|Maybe it%27s the whole %22bear%22 thing that%27s pulling us off track. What difference does it make to the problem if we change the word %22Bear%22 to %22coin%22%2C the status of %22male%22 and %22female%22 to %22heads%22 and %22tails%22 respectively%2C and link the idea of a bear being %22born%22 to a coin being tossed%3F%0D%0A%0D%0AMathematically and structurally it should make no difference%2C right%3F 50%2F50 is 50%2F50 whether it%27s male%2Ffemale or heads%2Ftails. You can toss a coin on Tuesday and a bear can be born on Tuesday. No matter the event that occurs on that day%2C it is still one day out of seven.%0D%0A%0D%0AThe whole bear story only serves as a distraction%2C I think. If not%2C please explain the difference. You keep indicating that there is a difference but I just don%27t see it. Here%27s the problem re-written word-for-word with the aforementioned substitutions%3A%0D%0A%0D%0A%22If%2C say%2C it is known that one of the coins is heads and was tossed on a Tuesday%2C what is the probability of the other coin to be also heads%3F%22%0D%0A%0D%0AWith these simple word substitutions would you still stand by the assertion that the probability is 48%25%3F%0D%0A%0D%0A
9|8|8|||||RE%3A Bear born on Tuesday problem|alexb||12:08:49|04/09/2011|%3E Maybe it%27s the whole %22bear%22 thing that%27s pulling us off track. What difference does it make to the problem if we change the word %22Bear%22 to %22coin%22%2C the status of %22male%22 and %22female%22 to %22heads%22 and %22tails%22 respectively%2C and link the idea of a bear being %22born%22 to a coin being tossed%3F%0D%0A%0D%0AI disagree with that analogy. It is important that%2C besides the mail%2Ffemale property%2C there is another one - birth day of the week. This is an intrinsic property of a bear%2Fcoin%3B being tossed on Tuesday is extrinsic to the experiment. This is what modifies the probabilities.%0D%0A%0D%0ATo make an analogy with coin tossing%2C the coins come in two weights - light and heavy - with the probabilities 1%2F7 and 6%2F7. It is known that one of the coins came up heads and was light%3B what is the probability that the other one came up heads%3F%0D%0A%0D%0AIf you wish you may build up a story. Coins are produced daily. All are identical%2C except for those that I made on Tuesdays. These ones are lighter. Two coins tossed. One happens to be lighter and show heads. What is the probability that the other one also shows heads%3F
10|1|0|||||RE%3A Bear born on Tuesday problem|Jon Hughes|1|17:49:52|05/02/2011|I also have a query on this problem%2C which may well be down to my own lack of understanding%2C but I think may hinge on the use of the words %27at least%27.%0D%0A%0D%0AIn the sample space supplied%2C there are 7 Mf%2C 7 fM and 13 Mm pairs%2C giving the probability of 13%2F27 that ONLY one bear is a male born on a Tuesday.%0D%0A%0D%0AHowever%2C if we are to accept at face value that %27at least one bear was born on a Tuesday%27%2C we open up the possibility that both were%2C meaning there is also 1 MM pair to add to this list%3A 14%2F27.%0D%0A%0D%0AIs this correct%3F%0D%0A
11|2|10|||||RE%3A Bear born on Tuesday problem|alexb||17:52:42|05/02/2011|If you look at the table then you%27ll see that at the intersection of the Tuesday column and the Tuesday row there is an event that two bears were born on a Tuesday.
12|3|11|||||RE%3A Bear born on Tuesday problem|Jon Hughes|1|22:11:38|05/03/2011|Understood%3B I think my mistake was in counting it twice.%0D%0A%0D%0AThe first list already contains the possibility that both bears were born on the Tuesday%2C so it shouldn%27t be counted again in the second list%3B hence 7%2C7%2C7%2C6 rather than 7%2C7%2C7%2C7.%0D%0A%0D%0AMany thanks.