Re: Combinometrics
In article [EMAIL PROTECTED], Jerry Dallal [EMAIL PROTECTED] wrote: Herman Rubin wrote: I also doubt whether learning to compute answers gives any insight into the concepts, except for those with good research potential, and even there it tends to confuse. It depends on what learning to compute means. (*I'm* saying this in repsonse to a comment from Prf. Rubin?!) Consider exp(i pi). I can compute it by using Euler's rule or by viewing it as the pi radians rotation of a rod of unit length in the imaginary plane. The second is not a means of computing, but of interpretation. Or consider the variance. I can compute it by using the desk calculator algorithm or by summing the squares of deviations. Knowing how to do it, and why, is not the same as the actual process of computing. I would go so far as to say that there is little, if any, point about computing the SAMPLE mean and the SAMPLE standard variance before understanding that of the population mean. Even population here is a bad term, as it implies that sampling without replacement is to be used, which is not the same as that of numerical functions of observations from arbitrary probability models. Even expectations should be done on a sample space, and it should be shown, or at least pointed out, that which equivalent formulation is used leads to the same results, including using the distribution as a particular one of these. If learning to compute means simply that one is given a formula--any formula--that is to be used without any thought of its origins, I agree. OTOH, thoughts about the method of computation can often lead to important insights. It is SOMETIMES the case that the procedure, not the method used to implement it, can do this. Setting up expectation on sample spaces makes additivity trivial; pointing out the equivalence of different representations makes expectation and variance of the binomial and hypergeometric quite easy, natural, and understandable. However, doing it using combinatorics provides no insight whatever, nor does using the cdf or pdf add much insight to anything about the concepts. Also, it is not necessary to introduce bivariate distributions to develop covariance, or its properties. Expectations of products are expectations, and simple algebra still works. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
David Heiser [EMAIL PROTECTED] wrote in message [EMAIL PROTECTED]">news:[EMAIL PROTECTED]... We seem to have a lot of recent questions involving combinations, and probabilities of combinations. I am puzzled. Are these concepts no longer taught as a fundamental starting point in stat? I remember all the urn problems and combinations of n taken m times, with and without replacements, the lot sampling problems, gaming problems, etc. These were all preliminary, early in the semester (fall). Now to see these questions popping up late in spring? Times may have changed, since the 1940's, and perhaps there is more important stuff to teach. Even if times hadn't changed, perhaps some of the posters aren't studying in the US, so their timetable may not match yours. (Right now it's late autumn where I am sitting.) Here in Australia, for example, the school year is the same as the calendar year - high schools will start in early February, universities will mostly start in early March (though it varies some from institution to institution). And not all posters are necessarily at university. However, I'd guess that many stats courses no longer do much combinatorial probability. Glen = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
Puzzle from last week: That said, there IS at least one natural application of such a sampling technique [random selection from equiprobable multisets], used in a major industry, where it saves millions of dollars a year. Answer: The casino/gaming industry... The wheel of fortune version of Crown and Anchor, which uses the [six multichoose 3 = 8 choose 3 = 56] multiset triples of six symbols on a wheel pays the casino far more handsomely than the chuckaluck version with three dice, which is in fact one of the more punter-friendly games. The player who wonders about the wheel game will notice that every possible combination is there (on the big wheels; there are smaller ones with some multisets omitted); but because it's every possible multiset, not every possible list, a higher proportion of outcomes are the doubles and triples, which (paying off at 2:1 for a double and 3:1 for a triple) at once look generous and actually lower the payout overall. If each number is covered equally, on a 1-1-1 outcome the house takes in $6 and pays out $6 ($3 returned bets + $3 winnings). On a 2-1, the house pays out $2 in returned bets and $3 winnings; and on a triple, only $4 in total. The wheel-of-fortune version keeps $0.125 for every $1 bet; the chuckaluck cage only $0.0787. (There are also smaller wheels which omit some of the 1-1-1 patterns (well, it wouldn't be fair to leave off the ones with _bigger_ prizes!) and do even better. Imagine the following scam, based on that psychology. The midway wheel operator has a couple accomplices in the crowd who do not hide the fact that they know him, but rather suggest that as friends they'd like a special game. Operator pretends that he's afraid of catching hell from the boss, but eventually gives in and explains to the other players that this means that all bets ride until there's a double or triple, and that he's not really meant to do this. Now, ladies and gentlemen, it's the same rules for everybody, so if you don't want to play keep your dollars in your pockets for this one game. When a shill loses he pleads for one more chance under the good rules unless one of the suckers is already doing it for him. And my, how the money rolls in... -Robert Dawson = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
Herman Rubin wrote: I also doubt whether learning to compute answers gives any insight into the concepts, except for those with good research potential, and even there it tends to confuse. It depends on what learning to compute means. (*I'm* saying this in repsonse to a comment from Prf. Rubin?!) Consider exp(i pi). I can compute it by using Euler's rule or by viewing it as the pi radians rotation of a rod of unit length in the imaginary plane. Or consider the variance. I can compute it by using the desk calculator algorithm or by summing the squares of deviations. If learning to compute means simply that one is given a formula--any formula--that is to be used without any thought of its origins, I agree. OTOH, thoughts about the method of computation can often lead to important insights. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
In article 9ctkri$fjvug$[EMAIL PROTECTED], Neville X. Elliven [EMAIL PROTECTED] wrote: David Heiser wrote: We seem to have a lot of recent questions involving combinations, and probabilities of combinations. I am puzzled. Are these concepts no longer taught as a fundamental starting point in stat? I haven't seen a Combinatorics course in a college class schedule in nearly twenty years, but combinations and their probabilities are still taught in Statistics courses [perhaps not with as much emphasis as previously]. Combinatorics used to be a standard topic in high school algebra. It is USED in probability calculations, which are USED in statistical calculations, but it is not either probability or statistics, no more than addition is. In fact, it is overdone; students have no problems with understanding equally likely, but have major problems with probability when this is not the case. I also doubt whether learning to compute answers gives any insight into the concepts, except for those with good research potential, and even there it tends to confuse. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
David Heiser wrote: We seem to have a lot of recent questions involving combinations, and probabilities of combinations. I am puzzled. Are these concepts no longer taught as a fundamental starting point in stat? I haven't seen a Combinatorics course in a college class schedule in nearly twenty years, but combinations and their probabilities are still taught in Statistics courses [perhaps not with as much emphasis as previously]. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
David Heiser wrote: We seem to have a lot of recent questions involving combinations, and probabilities of combinations. I've never seen multiset enumeration in elementary stats texts, perhaps because it is not very useful as a sampling model. While a multiset can certainly be the outcome of a sampling experiment, it is usually not natural to take a sample in which every multiset appears with the same probability, and so it is more useful to treat the (ordered) list of outcomes with repetition as the primitive model. It does turn up in thermodynamics and the discrete math courses taken by CS students. That said, there IS at least one natural application of such a sampling technique, used in a major industry, where it saves millions of dollars a year. Anybody know what I mean? I'll give the answer later! -Robert Dawson = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
