Path: news2.digex.net!digex.net!not-for-mail From: email@example.com (Michael P. Stein) Newsgroups: alt.revisionism Subject: Beastly poor math skills Date: 28 Dec 1996 18:26:54 -0500 Organization: Express Access Online Communications, Greenbelt, MD USA Lines: 270 Message-ID: <firstname.lastname@example.org> References:
<email@example.com> <firstname.lastname@example.org> <email@example.com> NNTP-Posting-Host: access1.digex.net Xref: news2.digex.net alt.revisionism:138773 In article <firstname.lastname@example.org>, anonymous wrote: >And now for a little exercise in probability and number theory -- Since Matt Giwer as usual turned tail and ran when asked to back up his claims of technical competence, it falls to me to serve as personal trainer and give a real workout to those flabby math muscles. >what are >the odds that accurately observed and reported statistical figures on such >diverse measures as numbers of people transported, numbers of deaths, >numbers of buildings, and so on turn out to be disproportionately multiples >of six? Here's my first cut at a solution (I'm ignoring a few >subtleties -- for example, I don't count 36 as being a multiple of >six twice, which would make the probabilities even lower): Not to mention a few unsubtlelties that make the probabilities much higher. OK, class, it's time for another lesson in "lies, damned lies, and statistics." The text for this course is Daniel Huff's excellent "How to Lie With Statistics." If you have not read it, by all means do so. Additional reading is any textbook on statistics and probability. >One in every six integers is a multiple of six. The probability that an >observed measure, whether rounded up to an integer or with the decimal point >removed, will be a multiple of six is 1/6. Not quite. That is only if the observed measure would have a normal tendency to be random. As an example, if someone mentions such-and-such every twenty-four hours, "24" cannot be counted as a number having any significance for this analysis. This is because the fact that a day _does_ have 24 hours makes it a "natural" figure which has a probability of 1 in this context, not a probability of 1 in 6. Second, one must ask whether it is truly an observed measure or an estimate. For psychological reasons, estimates have a bias towards multiples of 2, 3, and 5 and away from higher primes. >The probability that >two such measures will both be a multiple of 6 is 1/6^2 = 1/36. >(This is the same as the probability of throwing "snake eyes" with >a pair of six-sided die). In general, the probability that n measures >will all be multiples of 6 is 1/6^n. The probability that half of all >measures will be multiples of 6 is 1/6^(n/2). The probability >that m out of n measures will be multiples of six is 1/6^(n/(n/m)). > >In particular, the probability that five out of eleven >observed statistics will be multiples of six is > 1/6^(11/(11/5)) = 1 in 7,776. > >This is the same as the probability throwing "snake eyes" >twice in a row, then another snake eye after that, without interruption. > >Given an account or series of accounts with such statistics, >which is the more probable explanation: > >(a) The statistics were accurately observed and reported. > >(b) The Devil made the Nazis do it, in multiples of six. > >(c) The digit six and its multiples is a stylistic pattern used >throughout Judaic culture -- the six-sided Star of David, the "Number >of the Beast", Sorry to be a spoilsport here, but the "Number of the Beast" is a Christian idea, not Jewish. >and so on. The number six represents, not accurately reported >figures, but the use of this stylistic pattern in yet another Judaic >religious masterpiece. Did anyone in the class spot option (d) Anonymous here has made several really amazing errors in analysis of the text, statistical reasoning, and even simple counting? >And now for our sample of the precisely stated numerical >statistics in the "Myshkin letter": They are not, of course, precisely stated - not all of them. And as mentioned above, that's somewhat important, though not the biggest error that our anonymous friend makes. It's not even the second largest error. >1 six >>Every day, twelve thousand souls are being taken off... An estimate, not a precisely observed statistic. >4 not >>Four deportations of forty-five such train-loads move daily out of >>Hungary. Within twenty-six days all that area will have been >>deported. I see four, forty-five, and twenty-six. What is your fourth, math whiz? Three precise numbers. >1 not >>They are completely consumed in the >>ovens and leave no evidence behind. These are 95% of each >>transport. Estimate. >2 six >>The dead bodies are burned in specially made ovens. Each oven >>burns 12 bodies an hour. In February there were 36 ovens burning. I'll give Anonymous the benefit of the doubt and call this two precise numbers, though the first one is suspect. You see, if the writer had cast this as "Each oven burns a body every five minutes," we would have lost a six here - and it would look more suspiciously like an estimate. >1 not >>Information supplied us by a few eyewitnesses reveals that in >>February there were four disposal buildings. We have learned that >>more have been built since then. Precise number. >1 six >>This is the schedule of Auschwitz, from yesterday to the end; >>twelve thousand Jews - men, women and children, old men, infants, >>healthy and sick ones - are to be suffocated daily No. The 12,000 in this paragraph is the same as the 12,000 in the first paragraph. Counting the same six twice is not fair. >1 six (the Big One) >>... reaching now to six million Jews, were murdered. Also an estimate. After correcting for the duplication and miscount, we see 1/6^(9/(9/4)) = 1 in 1,296. If however we eliminate all the estimates, we only have 2 sixes out of six observations. That's only 1 in 36. Except.... Except of course it isn't. None of those computations are correct. Let's take a closer look at Mr. Number Theory's formula. Suppose there had been 24 observations, not nine. 1/6^(24/(24/4)) also = 1 in 1,296! And yet _that_ would give us exactly as many sixes as expected. Something is _very_ wrong with the formula he gave us. If we throw two dice the odds of double sixes are one in 36. But if we throw three dice, how do we compute the odds of getting at least two sixes? Intuitively, it should be higher, since if we miss a six with one die, we have one more shot. There are 216 possible combinations. 15 of them have exactly two sixes. (Dice 1 + 2, dice 1 + 3, dice 2 + 3; each multiplied by 5 for the non-six die which can vary from 1 to 5.) 1 has all 3 sixes. 16 of 216 is 1 in 13.5, not 1 in 36. The correct formula for the odds of precisely n sixes out of m tosses of a k-sided die is ((k-1)^(m-n) * (m!/n!(m-n)!))/k^m (m! is the symbol for m factorial, the product obtained by multiplying all the integers from 1 to m) Number theory, my you-know-what. In passing I shall mention that the correct standard should be the probability that _at least_ n of m measures would be multiples of 6, not _exactly_ n of m measures. Then of course there is the fact that in this case observed measures of one would usually not be counted in such an analysis, inflating the ratio of sixes to total measurements. But let's pretend the last problem doesn't exist, nor the problem of estimates being slightly nonrandom, and actually plug in numbers now that we are using the correct formula: ((6-1)^(9-4) * (9!/4!(9-4)!))/6^9 (5^5 * 126)/6^9 ~= 1 in 26. Not nearly so impressive, is it? And that's before we repeat for 5 through 9 sixes (because they have four sixes as well, their probabilities add into the _cumulative_ probability of _at least_ four sixes). >>And God who keeps alive the last remnant of Israel... > >Another theme common in Judaic religious literature, but I won't >count it. Invalid reasoning is a very common theme in "revisionist" literature. As Matt Giwer ought to say, revisionists are sneaky people, always relying on the inability of others to check and think critically about their "facts" and figures. We find that the chance of having exactly four sixes in nine rolls of a six-sided die is 1 in 26, and the odds of having _at least_ four would be even better - computation is left as an exercise for the reader. (Although I've cautioned why that die isn't really a random six-sided one.) That sure doesn't seem so significant, does it? And wait - it gets much, much worse. There is another Jewish tradition, something called gematria. It is a form of numerology where letters are mapped to numbers, and the words formed by the letters (or the number represented by one or more words) are analyzed for meaning. The rabbi at my congregation is a master at it. It is truly amazing what hidden meanings are encoded in _everything_. Of course that's much easier to do if you're allowed to fit a pattern on the data _after_ you get a look at it, rather than selecting the predicted pattern first and then seeing if the data fit. Here's a concrete example of what I mean. The odds of rolling three dice and getting the same number on each die is 6/216, or 1 in 36. But if you get to see the throw of the dice before you fit your pattern on it, you can do an after-the-fact analysis to announce some great significance for the number that actually came up, then claim that the because the odds were 1 in 216 of getting that _particular_ number, people should be suspicious that the dice were loaded, so to speak. Of course there are many other improbable patterns - e.g., two groups of three matching numbers in a set of 9. So paradoxically there is an _excellent_ chance that one will end up with some _specific_ pattern which is low-probability. Suppose the repeated number had been four? Four letters in the ineffable Name, four matriarchs. Five? Books of moses. Three? Patriarchs, number of pilgrimage festivals, number of categories of Jews (Kohen, Levi, Yisrael). Seven? Days of the week, number of days in the festival of Pesach (Passover) as observed in Israel, number of fat (or lean) years in Egypt. See how easy it is to "discover" some Jewish significance in a number when analysis is performed after the fact? But getting at least four identical results in nine throws of a six-sided die will happen 28.8% of the time, if you don't specify in advance _which_ number it is that should come up the same. That's why statisticians have a measure called a significance coefficient - when an apparently interesting pattern emerges from a set of observations, it is important to compute how likely a _similar_ (not necessarily identical) pattern is to emerge from a set of truly random numbers. The smaller the number of observations, the less significant the results are. On only nine observations, getting a _type_ of pattern which has a 28.8% shot of coming up is, in statistical terms, indistinguishable from random noise. And random noise, I'm afraid, is all that Mr. Anonymous's post turns out to be. "Revisionists are sneaky bastards, always relying on facts and figures." - Matt Giwer. "Figures don't lie, but liars figure." - Unknown -- Mike Stein The above represents the Absolute Truth. POB 10420 Therefore it cannot possibly be the official Arlington, VA 22210 position of my employer.
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