mm-3199 === Subject: probability question Can someone tell me how to figure out the probability for this? I don't just want the answer, I want to know how to get the answer. You have 6 dice. What is the probability of rolling all the dice and all six landing on the same number? I.e. 2,2,2,2,2,2 vs. What is the probability of rolling all six dice and all six landing on different numbers? I.e. 1,2,3,4,5,6 === Subject: Re: probability question > Can someone tell me how to figure out the probability for this? I don't just > want the answer, I want to know how to get the answer. > You have 6 dice. Probability is the ratio of successes to total outcomes. With 6 dice you have 6^6 (46656) possible outcomes. > What is the probability of rolling all the dice and all six landing on the > same number? I.e. 2,2,2,2,2,2 How many outcomes have all dice showing the same value? Probability would therefore be 6/46656. > vs. > What is the probability of rolling all six dice and all six landing on > different numbers? I.e. 1,2,3,4,5,6 Note, the total possible outcomes doesn't change. But the number of successfull outcomes does. If there are more than 6, the probability will be higher than getting all the same value. In your example, there is only one way to get 2,2,2,2,2,2 (and since there are 6 unique values, there are 6 total successful outcomes). For the example 1,2,3,4,5,6, there are no other numbers, but there are many ways to arrange them. Take the first die. Any outcome is considered a success, so for the first die, there are 6 successfull outcomes. Now for the second die, we can have any outcome EXCEPT the outcome that matches the first die. So there are only 5 possible successes for the second die. For the third die, same thing, anything is a success EXCEPT what shows on the first two dice. Continueing to the last die, we find that the diminishing number of successes leads to a final count of successes: 6*5*4*3*2*1 or 720. So probability of all dice different is 720/46656 or 120 times greater than getting all dice showing the same value. === Subject: Re: probability question this is what I got, does anyone know if this is correct? Probability of rolling all same digit: 1/46656 Probability of rolling all different digits: 5/324 (or 1/64.8) > Can someone tell me how to figure out the probability for this? I don't > just want the answer, I want to know how to get the answer. > You have 6 dice. > What is the probability of rolling all the dice and all six landing on the > same number? I.e. 2,2,2,2,2,2 > vs. > What is the probability of rolling all six dice and all six landing on > different numbers? I.e. 1,2,3,4,5,6 === Subject: Re: probability question | this is what I got, does anyone know if this is correct? | | Probability of rolling all same digit: 1/46656 I took the 1st question to mean, all 6 dice landing on the same number (any number), instead of a particular number. So, the 1st die has a 6 out of 6 chance to land on some number, and each die after that has a 1 out of 6 chance to land on that number: 6/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 = 1/6^5 = 1/7776 _______________________________________________________________Gerard S. | Probability of rolling all different digits: 5/324 (or 1/64.8) |> Can someone tell me how to figure out the probability for this? I don't |> just want the answer, I want to know how to get the answer. |> You have 6 dice. |> What is the probability of rolling all the dice and all six landing on the |> same number? I.e. 2,2,2,2,2,2 |> vs. |> What is the probability of rolling all six dice and all six landing on |> different numbers? I.e. 1,2,3,4,5,6 === Subject: Re: probability question Try looking up independent events! >Can someone tell me how to figure out the probability for this? I don't just >want the answer, I want to know how to get the answer. >You have 6 dice. >What is the probability of rolling all the dice and all six landing on the >same number? I.e. 2,2,2,2,2,2 >vs. >What is the probability of rolling all six dice and all six landing on >different numbers? I.e. 1,2,3,4,5,6 -- Casey === Subject: Re: probability question ??? > Try looking up independent events! >>Can someone tell me how to figure out the probability for this? I don't >>just >>want the answer, I want to know how to get the answer. >>You have 6 dice. >>What is the probability of rolling all the dice and all six landing on the >>same number? I.e. 2,2,2,2,2,2 >>vs. >>What is the probability of rolling all six dice and all six landing on >>different numbers? I.e. 1,2,3,4,5,6 > -- > Casey === Subject: where 2 start @ 4. Factor completely: 4x2 - 36y2 5. Factor completely: 3x2 - 2x - 8 6. Factor completely: 24x2 + 10x - 4 === Subject: Re: where 2 start @ > 4. Factor completely: > 4x2 - 36y2 I see (2x)^2 - (6y)^2 or 4 ( x^2 - (3y)^2 ) use the identity for difference of squares : u^2 - v^2 = (u+v)(u-v) > 5. Factor completely: > 3x2 - 2x - 8 this is a quadratic. when factorable they have the form (ax+b)(cx+d) for constant (and possibly negative) a,b,c,d ac=3 bd=-8 ad+bc=-2 start by guesing, if that fails find the roots. a=3 is probably a good starting point. > 6. Factor completely: > 24x2 + 10x - 4 as above. Bye. Jasen === Subject: Re: where 2 start @ > 4. Factor completely: > 4x2 - 36y2 > 5. Factor completely: > 3x2 - 2x - 8 > 6. Factor completely: > 24x2 + 10x - 4 Learn to use x^2 for x squared instead of x2. What are you doing? Having us do your homework? So you posted the first three problems in one newsgroup, alt.algebra.help and now you post the next three here and are you going to post another three in another newsgroup? Do you want to learn math or do you want your homework done for you? Learn some math. Memorise, a^2 - b^2 = (a - b)(a + b) and apply that frequently useful formula to your first, #4 problem. === Subject: Re: Using a mutliplacation table. >> Is using a table for homework cheating or more practice? > My problem is how would you effectively learn the multiplication tables if > you only use a table? You're going to need to know your multiplication facts > all your life, so why not memorize them? why not spend that time learning something else useful that can't be trivially derived? Bye. Jasen === Subject: Re: Using a mutliplacation table. > Is using a table for homework cheating or more practice? >> My problem is how would you effectively learn the multiplication tables >> if >> you only use a table? You're going to need to know your multiplication >> facts >> all your life, so why not memorize them? > why not spend that time learning something else useful that can't be > trivially derived? > Bye. > Jasen Such as? === Subject: Re: Using a mutliplacation table. <4f85.4428ddf4.c9f1a@clunker.homenet> Is using a table for homework cheating or more practice? > >> My problem is how would you effectively learn the multiplication tables >> if >> you only use a table? You're going to need to know your multiplication >> facts >> all your life, so why not memorize them? > why not spend that time learning something else useful that can't be > trivially derived? > Bye. > Jasen > Such as? For my niece it is percent for now. She has the concept of percent down but she gets hung up on the math. (I get hung up on the math) === Subject: Re: Using a mutliplacation table. >>> Is using a table for homework cheating or more practice? >>>> My problem is how would you effectively learn the multiplication >>> tables >>> if >>> you only use a table? You're going to need to know your multiplication >>> facts >>> all your life, so why not memorize them? >> why not spend that time learning something else useful that can't be >> trivially derived? >> Bye. >> Jasen >> Such as? > For my niece it is percent for now. She has the concept of percent > down but she gets hung up on the math. (I get hung up on the math) My point is that everyone should know the basic multiplication tables. It will come in handy when she gets to, say, algebra. Dave === Subject: Re: Sum(n!/(2n+1)!) from n=1 to infinity (Sorry for repost) >> (Sorry for the repost I thought it better to repost to all newsgroups >> under one than to repost it seperately in other, better of two evils imho). >> >> Hi I was (and still am slightly) stuck on a problem someone posed me >> (not a lecturer thank god it's taken me the last 7 hours and a nights >> sleep inbetween), namely to determine what was the limit of >> >> Sum(n!/(2n+1)!) from n=1 to infinity (1) >> >> If I were doing this by hand, I'd try setting >> f(x) = sum_{n=0}^infty n!/(2n+1)! x^n, >> so that >> x f(x^2) = sum n!/(2n+1)! x^(2n+1). >> Differentiating, >> f(x^2) + 2x^2 f'(x^2) = sum n!/(2n)! x^(2n) >> = 1 + 1/2 sum_{n >= 1} (n-1)!/(2n-1)! x^(2n) >> = 1 + 1/2 x^2 sum_{n >= 0} n!/(2n+1)! x^(2n) >> = 1 + 1/2 x^2 f(x^2). >> Thus y = f(x) satisfies the differential equation >> y + 2xy' = 1 + 1/2 x y. >> The solution of this isn't QUITE straightforward, but it gets you your >> erf. (Muliply left and right by x^(-1/2) to get an exact integrating >> factor.) >Someone has asked me for details of the solution of the ODE, > y + 2xy' = 1 + 1/2 x y. >Multiply both sides by x^(-1/2), which is an integrating factor: > (2x^(1/2)y)' = x^(-1/2) + 1/4 (2x^(1/2)y). >Set w = 2x^(1/2)y, thus > w' = x^(-1/2) + 1/4 w. >Thus > w' - 1/4 w = x^(-1/2). >The LHS now has e^(-1/4 x) as an integrating factor, > (e^(-1/4 x) w)' = x^(-1/2) e^(-1/4 x), >and the RHS is easy to integrate (substitute x = 4s^2, so > int x^(-1/2) e^(-1/4 x) dx = int 1/2 s^(-1) e^(-s^2) 8s ds > = 4 int e^(-s^2) ds > = 4 erf(s) = 4 erf(sqrt(x)/2), >depending on your definition of erf. Thus > e^(-1/4 x)w = C + 4 erf(x^(1/2)/2), > w = C e^(x/4) + 4 e^(x/4) erf(x^(1/2)/2). >We finally get y = 1/2 x^(-1/2) w. Things are a bit simpler if we introduce a slightly different sum. Define oo --- n! 2n+1 f(x) = > ------- x [1] --- (2n+1)! n=0 Note that f(0) = 0. Differentiate both sides, oo --- n! 2n f'(x) = > ----- x --- (2n)! n=0 oo --- x (n-1)! 2n-1 = 1 + > - ------- x --- 2 (2n-1)! n=1 oo x --- n! 2n+1 = 1 + - > ------- x 2 --- (2n+1)! n=0 x = 1 + - f(x) [2] 2 Rearrange [2] and multiply by an integrating factor of e^{-x^2/4} and we get -x^2/4 -x^2/4 ( e f(x) )' = e [3] -x^2/4 e f(x) = sqrt(pi) erf(x/2) Since f(1) is 1 greater than the desired sum, we get that the sum is 1/4 e sqrt(pi) erf(1/2) - 1 = 0.18459307293865315132 This uses the normalization of erf where erf(oo) = 1. Rob Johnson take out the trash before replying === Subject: feebly continuous mappings continuous and feeble open mappings? I need to prove that a function is feebly continuous if and only if it preserves dense subsets. The fact was known already in 1960, but I can not proove it myself. Natalia === Subject: Re: Anyone care to check my homework.? >It is my niece's homework really. I want to make sure I can do them >before I show her how. >I have been out of school for almost 30 years. >I catch myself using the excuse I always got... They didn't have >that when I was in school. >1-3 http://tinypic.com/sdz3ht.jpg >4-7 http://tinypic.com/sdzivo.jpg >8-11 http://tinypic.com/sdz80x.jpg >12-14 http://tinypic.com/sdz9n6.jpg >15-17 http://tinypic.com/sdzbt4.jpg we don't grade papers! === Subject: Re: Anyone care to check my homework.? We? You have a mouse in your pocket? I have my answers from here. >It is my niece's homework really. I want to make sure I can do them >before I show her how. >I have been out of school for almost 30 years. >I catch myself using the excuse I always got... They didn't have >that when I was in school. >1-3 http://tinypic.com/sdz3ht.jpg >4-7 http://tinypic.com/sdzivo.jpg >8-11 http://tinypic.com/sdz80x.jpg >12-14 http://tinypic.com/sdz9n6.jpg >15-17 http://tinypic.com/sdzbt4.jpg > we don't grade papers! === Subject: Re: Only 45% of the students were prepared for math > Los Angeles Times Mar 15, 2006 Page B9 > CSU Freshmen Face Challenges; Only 45% of the students were prepared > for math and English studies at college level, report says > Author(s): Cynthia H. Cho > Document URL: > http://proquest.umi.com/pqdweb?did=1003262501&Fmt=3&clientId=16778&RQT=309&V N ame=PQD > Wednesday in California about the California State University system > described the number of students who benefited from English and math Those needing remedial courses should be required to take them at a community college, and NOT at a state university that is supported by tax money. These students should not have been accepted in the first place. And I doubt that the problem is restricted to just math and English. I strongly suspect that most of these students simply lacked the dedication and intellectual maturity required at any college in any subject. Too many students go to college just to party and have a good time. Maybe if they had to pay for their own education, rather than just partying at mommy's and daddy's expense, they would actually dedicate themselves to LEARNING. === Subject: Re: Only 45% of the students were prepared for math <25o822pt3oh9p6jjqq70ietqgasccq929h@4ax.com> On 24 Mar 2006 21:10:22 -0500, hrubin@odds.stat.purdue.edu (Herman >>One needs very little of a language. Scientists have >>demonstrated that little vocabulary is learned before >>the idea of grammatical structure is managed; there >>has been an argument, with data to back it, that >>children earlier than one year, with zero vocabulary, >>can comprehend grammatical ideas. >Herman, you do this constantly. What scientists? How many >studies? Have these studies been replicated? Where is this >data? For heaven's sake, what parents have even allowed >their children of less than a year old to be involved in this kind >of foolishness? > He seems to be arguing that because children who are a year old learn > correct grammar by absorption without explanation, that they could > similarly learn mathematical thinking by absorption without > explanation. > The flaw is that a two year old does not in fact understand any of > the rules that he follows; he just follows them, very concretely > (though the result looks like an abstraction to those who are thinking > abstractly). which he couldn't quite recall. The idea of young children understand grammatical ideas before vocabulary has to have come from someone who is linguistically challenged. The grammatical structure of Romance languages (French, Spanish) and Germanic languages (German) and their derivatives from Latin, such as English, are vastly different. Children who are less than 6 years old should have no difficulty being multi lingual in English, French, Spanish, German, and even Chinese. The older one gets, the more difficult it is to learn. I was told by a Berlitz teacher that I tried to THINK too much about the grammar of Spanish (when I already learned those in French, German, and English). I was no match for any 6 year old. In fact, when I was 6 years old, I was multi-lingual in four dialects of Chinese -- each of which is as different as German, French, and English. Young children have a completely different way of learning a language, by imitation of rules rather than abstraction of rules. Vocabulary certainly precedes grammar. At Yale, there is a tall science building called the Kline Tower. The mother of a German kid mused when her son couldn't understand why the tower was Klein when the word means small in German. :-) The kid had no trouble speaking English and German fluently. I had the hardest time learning why every inanimate object is male, female, or neuter in German, and how to look for the separable prefixes and suffixes that may be a page or two away. Kids NEVER had that kind of problem in learning German. Scientist are often blinded by their own prejudices and ignorance in conjecturing and testing the untestible, as in less than 1-yr olds. But often they get GRANTS to do the silliness, if they can BS enough pages in application for the grant. -- Bob. === Subject: Re: Only 45% of the students were prepared for math >>On 24 Mar 2006 21:10:22 -0500, hrubin@odds.stat.purdue.edu (Herman >>>One needs very little of a language. Scientists have >>>demonstrated that little vocabulary is learned before >>>the idea of grammatical structure is managed; there >>>has been an argument, with data to back it, that >>>children earlier than one year, with zero vocabulary, >>>can comprehend grammatical ideas. >>Herman, you do this constantly. What scientists? How many >>studies? Have these studies been replicated? Where is this >>data? For heaven's sake, what parents have even allowed >>their children of less than a year old to be involved in this kind >>of foolishness? >> He seems to be arguing that because children who are a year old learn >> correct grammar by absorption without explanation, that they could >> similarly learn mathematical thinking by absorption without >> explanation. >> The flaw is that a two year old does not in fact understand any of >> the rules that he follows; he just follows them, very concretely >> (though the result looks like an abstraction to those who are thinking >> abstractly). >which he couldn't quite recall. Herman thinks it said. It said that the infants had shown that they had learned grammatical rules by some sort of pattern identification, not that the could comprehend grammatical ideas which suggests that they consciously could think about those rules. http://www.psych.nyu.edu/gary/marcusArticles/marcus%20et%20al%201999%20scien ce.pdf and here are some of the multitude of commentaries on the interpretation of the results. http://lcnl.wisc.edu/people/marks/courses/lang&mind/6marcusLetters.html http://cnl.psych.cornell.edu/abstracts/transfer-learning.html http://www.psych.nyu.edu/gary/marcusArticles/marcus%202000%20cdps.pdf Here is another reporting on experiments with somewhat older infants. >The idea of young children understand grammatical ideas before >vocabulary has to have come from someone who is linguistically >challenged. They can *recognize* simple grammatical rules at a time fairly close to when they start to *recognize* that particular words have meaning. The issue is whether pattern recognition is understanding. I didn't even know what a differential equation was, when I tutored a kid who was reading a textbook section on solving 2nd order linear differential equations - I recognized the pattern of the quadratic formula in the method of solution. That doesn't mean that I had any clue as to why the quadratic formula could be used, or even what solution meant other than getting the right answer to the problem, but having recognized the pattern that the other kid had missed, HE was able to better understand. Thus clearly pattern recognition is a useful tool in abstraction, but I don't think it constitutes understanding. >Young children have a completely different way of learning a >language, by imitation of rules rather than abstraction of rules. Tough call because Herman uses a vague definition of abstraction. Clearly They must DO some kind of abstract pattern recognition in order to be able to imitate rules, since that seems to be how kids learn which rules to imitate. >Vocabulary certainly precedes grammar. >At Yale, there is a tall science building called the Kline Tower. >The mother of a German kid mused when her son couldn't >understand why the tower was Klein when the word means >small in German. :-) And yet he understood that Klein was modifying Tower and not vice versa (there are languages where the adjective comes after the noun). >Scientist are often blinded by their own prejudices and ignorance >in conjecturing and testing the untestible, as in less than 1-yr olds. >But often they get GRANTS to do the silliness, if they can BS >enough pages in application for the grant. But the research wasn't silly - it just wasn't testing what Herman seems to think it was testing. lojbab === Subject: Re: Only 45% of the students were prepared for math <4n0422lefgejing6pmven5fvl0plri97gh@4ax.com> <4m9d22thbas0cqqc14uv86eoual2d05um5@4ax.com> Every time I see this endless thread, I think, ... and the other 65% > have no clue! > Are you suggesting that 65 + 45 = 100? > While others wondered why: Only 45% of the students were prepared for > math > Killfiler seemed to have something much more interesting and > substantial > than you had to say. I found fishfry's tidbit very amusing. Correcting his/her math only spoils the joke. -- Patricia Burns (Just one s) === Subject: Re: Only 45% of the students were prepared for math <4n0422lefgejing6pmven5fvl0plri97gh@4ax.com> <4m9d22thbas0cqqc14uv86eoual2d05um5@4ax.com> <4428B73F.359906E3@burns.net > Every time I see this endless thread, I think, ... and the other 65% > have no clue! > Are you suggesting that 65 + 45 = 100? > While others wondered why: Only 45% of the students were prepared for > math > Killfiler seemed to have something much more interesting and > substantial > than you had to say. > I found fishfry's tidbit very amusing. Correcting his/her math only > spoils the joke. Sorry about missing your k.12.chat mentality of a joke. I've seen it too many times where it was no joke. Since I don't know fishfry from fish 'n chips, I naturally assumed that s/he was one of the k.12 or pre-kintergarden teachers who have difficulty with double digit arithmetic. I have enough trouble with the college students and teachers making the same 'rithmetic errors. :) They are the graduates of the kind of classes killfiler and I talked about -- everyone expects, and most get A's for paying the tuition -- even attendance is not required, as in k.12. I don't find anything amusing about finding tree stumps in class disguised as college students. -- Bob.