* As it happened, only seven people matched five numbers that day, so each of those luckies shared a prize of over $80,000. But the scarcity of those winners seems to have been just bad luck, and not something you could fairly have anticipated when computing the expected value of a ticket in advance.
* Given the popularity of Cash WinFall, this is actually somewhat surprising; there was about a 10% chance per roll-down that somebody would win the jackpot, so it should have happened four or five times. That it happened only once was, as far as I can tell, plain bad luck—or, if you like, good luck for the people counting on those lesser roll-down prizes.
* As long as we ignore all the prize money that doesn’t come from the roll-down; but as we’ve seen, that money doesn’t amount to much.
* Still ignoring the money that doesn’t come from the jackpot fund.
* Of Cramer’s rule, for all the linear algebra fans in the house.
* Actually, it’s not totally clear to me that he was actually “Buffon” at the time of his Academy presentation; his father, who had bought the title of comte de Buffon in the first place, had mismanaged his business and had to sell the Buffon holdings, and meanwhile had remarried a twenty-two-year-old; Georges-Louis sued and apparently managed to divert his mother’s childless uncle’s fortune directly to himself, allowing him to buy back both land and title.
* You might complain that since the needle is exactly as long as the slat is wide, it is possible that the needle touches two cracks. But this requires that the needle span the slat exactly; it is possible, but the probability that it happens is 0, and we can safely ignore it.
* Customarily pronounced “you-tills,” but in my experience it’s much more fun to say “yoodles.”
* Although I’ve heard at least one economist argue that since a certain amount of future happiness is worth less than the same amount of happiness now, the value of eternal joy in the bosom of Abraham is actually finite.
* Though remember from chapter 2 that divergent series aren’t just the ones which shoot off to infinity; they also include those which fail to settle down in other ways, like Grandi’s series 1 − 1 + 1 − 1 + . . .
* Warning: great dangers await when using this kind of intuitive argument with infinite sums. It’s okay in the case at hand, but wildly wrong for knottier infinite sums, especially those with both positive and negative terms.
* Although, as Karl Menger—Abraham Wald’s PhD advisor—pointed out in 1934, there are variants of the St. Petersburg game so generous that even Bernoulli’s logarithmic players would seem to be bound to pay arbitrarily many ducats to play. What if the kth prize is 22^k ducats?
* Indeed, most people would say the utility curve does not even literally exist, as such—it should be thought of a loose guideline, not as a real thing with a precise shape we haven’t yet measured exactly.
* Lebowitz wrote in her book Social Studies, “Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.” I claim this example shows there is mathematics in Lebowitz’s life, whether she refers to it as such or not!
* The same Oskar Morgenstern who got Abraham Wald out of pure math and eventually out of occupied Austria.
* I have never even seen an urn, but it’s some kind of iron law of probability theory that if randomly colored balls are to be chosen, it must be an urn that holds them.
* Analysts like Nassim Nicholas Taleb argue, persuasively in my opinion, that it’s a fatal error to assign numerical probabilities to rare financial events at all.
* Of course there’s ample reason to believe that some people inside the banks knew their investments were pretty likely to founder and that they lied about this; the point is that even when bankers are honest the incentives push them toward taking stupid risks at the public’s eventual expense.
* 366 if you count leap days, but we’re not going for precision here.
* The first person in the pair can be any of the 30 people, and the second any of the 29 who remain, giving 30 × 29 choices; but this counts each pair twice, since it counts {Ernie, Bert} and {Bert, Ernie} separately; so the right number of pairs is (30 × 29)/2 = 435.
* Unless you’ve heard of a googolplex. Now that is a big number, boy howdy.
* Or at least they looked like certain kinds of optical representations of the things they were paintings of, which over the years we’ve come to think of as realistic; what counts as “realism” has been the subject of hot contention among art critics for about as long as there’s been art criticism.
* Anachronistic, okay, but just go with it.
* But if the lines containing R are all horizontal, and the lines containing P are all vertical, what is the line through R and P? It is a line we haven’t drawn, the line at infinity, which contains all the points at infinity and none of the points of the Euclidean plane.
* To be fair, there is another sense in which the Fano plane really does look like more traditional geometry. Descartes taught us how to think of points on the plane as pairs of coordinates x and y, which are real numbers; if you use Descartes’s construction but draw your coordinates from number systems other than the real numbers, you get other geometries. If you do Cartesian geometry using the Boolean number system beloved of computer scientists, which has only two numbers, the bits 0 and 1, you get the Fano plane. That’s a beautiful story, but it’s not the story we’re telling just now. See the endnotes for a little more of it.
* And every signal is noisy, to some degree or another.
* Ian Hacking’s The Emergence of Probability covers the story superbly.
* For the technical sticklers, what I’m describing here is actually the dual of the usual Hamming code; in this case, it’s an example of a punctured Hadamard code.
* If the original codeword is 0000000, then the version with one bit messed up has six 0s and only one 1, making the receiver quite confident that 0000000 was the intended signal.
* If you haven’t thought about this before, you have probably found that the argument in this paragraph is hard to follow. The reason it’s hard to follow is that you can’t get an argument of this kind into your brain by sitting and reading about it—you have to get a pen out and try to write down a set of four points which contains two different lines in the Fano plane, and then fail to do that, and then understand why you failed. There is no other way. I encourage you to write directly in the book, if it’s not borrowed from the library or displayed on a screen.
* For the experts: that Hamming distance satisfies the triangle inequality.
* Not the same thing!
* I would like to think the fact that bebop is Ro for “elastic” is an undiscovered fragment of the secret history of jazz, but it’s probably just a coincidence.
* According to the FAQ at lojban.org, the number of people who can speak Lojban conversationally “ranges beyond what can be counted on the fingers of one hand,” which in this business is indeed pretty good.
* To be more precise, a sphere is the set of points at distance exactly 1 from the center; the space described here, a filled-in sphere, is usually called a ball.
* Which is to say, at distance either 0 or 1, since Hamming distances, unlike the usual distances in geometry, have to be whole numbers.
* We do know, though, that the atoms in the solid forms of aluminum, copper, gold, iridium, lead, nickel, platinum, and silver arrange themselves in face-centered cubic form. One more example of a mathematical theory finding applications its creators could not have contemplated.
* Though in contexts where signals are modeled as sequences of real numbers, not sequences of 0s and 1s, the sphere-packing problem is precisely what’s needed to design good error-correcting codes.
* Cohn works at Microsoft Research, which is in a way a continuation of the Bell Labs model of pure math supported by high-tech industry, hopefully to the benefit of both.
* Yet another great story too long and twisty to wander into here,
but see Mark Ronan’s Symmetry and the Monster.
* What’s the point, when Shannon proved that a totally random choice of code should work just as well? Yes, in a sense, but his theorem in its strongest form requires that code words be allowed to get arbitrarily long. In a case like this, where the code words are fixed to have length 48, you can beat the random code with a little extra care, and this is exactly what Denniston did.
* In math terms, this is because Denniston’s list of tickets forms what’s called a Steiner system. Added in press: In January 2014, Peter Keevash, a young mathematician at Oxford, announced a major breakthrough, proving the existence of more or less all possible Steiner systems that mathematicians had wondered about.
* I am not making this argument up; if you want to see it pushed through in full, see Gary Becker and Kevin Murphy’s theory of rational addiction.
* He apologizes in the introduction for the omission of foreigners, remarking, “I should have especially liked to investigate the biographies of Italians and Jews, both of whom appear to be rich in families of high intellectual breeds.”
* Technical but important note: When Galton says “necessary,” he is making use of the biological fact that the distribution of human height is roughly the same from generation to generation. It’s theoretically possible for there to be no regression, but this would force an increase in variation, so that each generation would have more gigantic giants and more diminutive pipsqueaks.
* It’s hard to understand how Secrist, who was familiar with Galton’s work on human height, managed to convince himself that regression to the mean was found only in variables under human control. When a theory really has got your brain in its grip, contradictory evidence—even evidence you already know—sometimes becomes invisible.
* These cases are complicated by the fact that novelists and musicians tend to get better with more practice. F. Scott Fitzgerald’s second novel (can you even name it?) is pretty bad compared to his debut, This Side of Paradise, but when his style matured he turned out to have a little bit left in the tank.
* This fact, along with its interpretation, comes from Brian Burke at Advanced NFL Stats, whose clear exposition and rigorous attention to statistical good sense should be a model for all serious sports analysts.
* Actually, the overall home run rate appears to dip slightly in the second half; but this may be because late-season call-ups are getting more at bats. In a data set consisting of elite home run hitters, the second-half home run rate and the first-half home run rate were the same (J. McCollum and M. Jaiclin, Baseball Research Journal, Fall 2010).
* The authors do gesture at the existence of regression: “While this phenomenon could merely be a regression towards the mean, we conclude that increasing the fibre intake does have a genuine physiological action in slowing fast transit times and accelerating slow transit times in patients with diverticular disease.” Where this conclusion comes from, apart from faith in bran, is hard to say.
* Or at least reinvented it: the astronomer John Herschel constructed a sort of scatterplot in 1833 to study the orbits of binary stars. This isn’t the same Herschel who discovered Uranus, by the way; that was his dad, William Herschel. Notable Englishmen and their notable relatives!
* What Nightingale actually called the coxcomb was the booklet containing the graph, not the graph itself, but everybody calls it the coxcomb and it’s too late to change it now.
* The 1.08 is to make the average heights of the mothers roughly match those of the fathers, so that male and female height are measured on the same scale.
* Isopleths go back even further than this. The first ones we know about were isobaths (curves of constant depth) drawn on maps of rivers and harbors, which go back at least as far as 1584; but Halley seems to have invented the technique independently, and certainly popularized it.
* You could also make a case for curves of exponential growth and decay, which are just as ubiquitous as conic sections.
* Why they are called quadrics as opposed to quadratics is a nomenclatural mystery I have not managed to penetrate.
* There are actually a few extra cases, like the curve with the equation xy = 0, which is a pair of lines crossing at the point (0,0); these are considered “degenerate” and we will not speak of them here.
* For all his enthusiasm for data, though, Bertillon blew it in the biggest case he ever handled; he helped convict Alfred Dreyfus of treason with a bogus “geometric proof” that a letter offering to sell French military documents was written in Dreyfus’s handwriting. See L. Schneps and C. Schneps, Math on Trial, for a full account of the case and Bertillon’s unfortunate involvement.
* That’s how Fosdick tells the story, at any rate, in “The Passing of the Bertillon System of Identification.” As with any famous crime of yesteryear, there’s a huge accretion of uncertainty and conspiracy theory around the Mona Lisa theft, and other sources tell different stories about the role of the fingerprints.
* Readers of a certain age may enjoy knowing that the Parsons who invented the Parsons code was the father of Alan Parsons, who recorded “Eye in the Sky.”
* Okay, it’s not literally just a matter of correlations between pairs of pixels, but it does come down to the amount of information (in Shannon’s sense) conveyed by an image.
* Dad of Egon Pearson, who battled with R. A. Fisher in an earlier chapter.
* Though perhaps it’s best not to complain too loudly about the incorrect use of exponential to mean simply “fast”—I recently saw a sportswriter, who had no doubt been scolded at some point about exponential, refer to sprinter Usain Bolt’s “astonishing, logarithmic rise in speed,” which is even worse.
* Except insofar as the whole stock market tends to move in concert, of course.
* Technical note for those who care: in fact, this is the two-dimensional projection provided by a principal component analysis on the poll answers, so the uncorrelatedness of the two axes is automatic. The interpretation of the axes is my own. This example is meant merely to illustrate a point about correlation, and should not under any circumstances be taken as actual social science!
* Though see the work of Judea Pearl at UCLA, whose work is at the center of the most notable contemporary attack on the problem of formalizing causality.
* All numbers in this example made up with no regard for plausibility.
* Or “people of your preferred gender, if any,” obviously.
* Added in press: A CNN/ORC poll in May 2013 found that 43% favored the ACA, while 35% said it was too liberal and 16% said it wasn’t liberal enough.
* People argue to this day about whether Perot took more votes from Bush or from Clinton, or whether the Perot voters would have just sat it out rather than vote for either of the major-party candidates.
* On May 15, 1805, Massachusetts outlawed cropping, along with branding, whipping, and the pillory, as punishments for counterfeiting money; if those punishments had been understood to be forbidden by the Eighth Amendment at the time, the state law would not have been necessary (A Historical Account of Massachusetts Currency, by Joseph Barlow Felt, p. 214). Scalia’s concession, by the way, doesn’t reflect his current thinking: in a 2013 interview with New York magazine, he said he now believes the Constitution is A-OK with flogging, and presumably he feels the same way about cropping.
* Since 2002, the number has risen to seventeen.
* This is not precisely Scalia’s computation; Scalia didn’t go so far as to assert that the no-death-penalty states thought execution of mentally retarded criminals no worse than execution in general. Rather, his argument amounts to the claim that we have no information about their opinions in this matter, so we shouldn’t count these states in our tally.
* Yes, I, too, know that one guy who thought both Gore and Bush were tools of the capitalist overlords and it didn’t make a difference who won. I am not talking about that guy.
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 51
