God Plays Dice

Last week I wrote about confusing coffee pricing: Wawa, a Philadelphia-area convenience store chain, charges 1ドル.25 for 32 ounces of coffee and 2ドル.99 for 64 ounces. 2ドル.99 is more than twice 1ドル.25. Various commenters pointed out other counterintuitive pricing (train or airline fares that don't obey the triangle inequality, for example). Paul Soldera pointed out in a comment that the reason for this may just be that there aren't that many mathematicians out there, and 3ドル for 64 ounces of coffee sounds like a bargain to most people.

Paul Soldera may be right -- but I discovered another candidate explanation for this pricing today. Namely, I took a closer look at a sign (at a different Wawa from the one I normally go to), and it said that the 64-ounce "includes supplies". In other words, they're not selling this as a giant cup of coffee for one person to drink, but as something from which you can pour multiple cups for multiple people. Thus, they provide the cups, and perhaps other coffee paraphernalia as well.

From Kitchen Table Math, I found out that there is a textbook entitled "New York Geometry". It is, unsurprisingly, a textbook for high school geometry to be used in the state of New York. The state of New York has something called the Regents Examinations which are, roughly speaking, standardized final exams in certain subjects. (Not being too familiar with the system, I don't want to say more.)

But is geometry really so different in New York than in other states that it needs its own special book? If it is, I can't tell from the table of contents of the text; it sounds like standard high school geometry.

From The Dilbert Blog (by Scott Adams), on the fact that most people are not racists but most people say they know racists, and how it affects the upcoming presidential election:

The other inference is something I call math. If there are ten friends, and only one is a racist, then it is true that 90 percent are not racists while everyone knows someone who is. It’s that one guy.

This isn't quite true -- if the average person has ten friends, and ten percent of people are racist, then the average person has one racist friend. But even if friends are randomly distributed, the probability that I have no racist friends is (0.9)¹⁰ or about 35 percent. And friends aren't randomly distributed. Most people tend to have people like themselves as friends. So the probability of having no racist friends is higher.

Still, it's a good point; we are not our friends, and our friends can believe different things than we do, and that's not a problem. (Incidentally, Barack Obama is not Jeremiah Wright.)

From Lydia Millet's novel Oh Pure And Radiant Heart, in which Robert Oppenheimer, Enrico Fermi and Leo Szilard, three of the chief minds behind the Manhattan Project, find themselves in contemporary America (p. 290):

Scientists at the Atomic Energy Commission took advantage of the testing in the Marshall Islands to study the effects of radiation on people.
In 1956, at an AEC meeting, one official admitted that Rongelap was the most contaminated place on earth. He said of the Marshall Islanders, reportedly without irony, "While it is true that these people do not live, I would say, the way Westerners do — civilized people — it is nevertheless true that they are more like us than mice.

To my ear, something about this last sentence is ambiguous -- and I suspect that a mathematician is more likely to spot this ambiguity than an average person. Let's assume that degrees of civilizedness fall on a scale from 0 to 1, with mice at 0 and Westerners at 1. Say that we have a magical civilizedness-measuring meter, and the Marshall Islanders fall at 1/3. Is the scientist's statement true?

If the scientist is saying that the distance in some abstract civilizedness-space (here caricatured by the unit interval) from Westerners to Marshall Islanders is less than the distance from Westerners to mice, then yes. The last clause could be rephrased as "it is nevertheless true that they are more like us than mice are like us." But if the scientist is saying that the distance from Westerners to Marshall Islanders is less than the distance from Marshall Islanders to mice, then it's not true if the islanders fall at 1/3; to force this interpretation, the original sentence could be rephrased as "it is nevertheless true that they are more like us than they are like mice." (I make no claim that these are the most elegant possible rephrasings, just that they clear up the ambiguity.)

Of course, in this particular case, I would claim the scientist intended the second interpretation; regardless of what one thinks about how civilized various groups of humans are, it is obvious that all such groups are more civilized than mice. (I apologize to fans of the Hitchhiker's Guide series.) So there is no need to even make the statement under the first interpretation! There is not much point in telling people something they already know.

Also, this shouldn't need saying, but the value 1/3 above is entirely hypothetical, and I do not mean to make any statements about the civilizedness of actual groups of life forms.

Study Suggests Math Teachers Scrap Balls and Slices, from today's New York Times.

The Times article is about a study reported on in today's issue of Science (Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler1. The Advantage of Abstract Examples in Learning Math. Science 25 April 2008: Vol. 320. no. 5875, pp. 454 - 455). Researchers taught the idea of the group Z₃ to some students who weren't familiar with it; some learned it "abstractly" (the elements of the group were represented as funny-looking symbols) and some learned it "concretely" (by considering the slices in a pizza with three slices, or thirds of a measuring cup, or tennis balls in a three-ball can). It seems that the ones who learned the "abstract" version more easily picked up the rules of yet another "concrete" version (a children's game) than those who learned the original "concrete" version.

The Science authors claim that this is because "Compared with concrete instantiations, generic instantiations present minimal extraneous information and hence represent mathematical concepts in a manner close to the abstract rules themselves." This seems like the whole point of mathematics -- a lot of what we do as mathematicians is to strip away extraneous details of a problem while retaining those that are actually significant. If you learn about fractions by thinking about slices of pizza, perhaps you will always think that fractions are about pizza. And then whenever you hear about them, you'll think "where's lunch"?

In tonight's Pennsylvania primary, the structure of the delegate allocation heavily favors to Obama, according to cnn.com video coverage.

The argument is the following: there are 55 at-large delegates which are assigned proportionally to the popular vote in the entire state. There are also 103 delegates divided up among the 19 Congressional districts, with more heavily Democratic districts receiving more votes. (For example, the 2nd district -- mine -- gets nine delegates, which I think is the most of any district nationwide. That's basically the western half of Philadelphia.) The 9th district gets the fewest, with three; numbers for other districts are here.

Now, the formula that assigns the delegates (I can't find it right now) basically says that the number of delegates that a district gets is proportional to the number of Democratic votes in the last few elections.

So assuming turnout is stable, the outcome really isn't any different than it would be if all the delegates were assigned "at large" -- up to rounding errors from the fact that delegates are quantized, but I don't believe the rounding errors break consistently one way or the other. (Roundinf error often do.)

For a small example, consider a hypothetical state with two districts. The first district historically has a turnout of 45,000 Democrats and gets three delegates; the second district historically has a turnout of 105,000 Democrats and gets seven delegates. In addition there are five at-large delegates.

Now say the turnouts in the election are the same; and 65% of Democrats in the first district vote for Clinton, and 65% of Democrats second district vote for Obama. So the first district breaks 29,250 to 15,750 for Clinton, and the second 68,250 to 36,750 for Obama. The state as a whole goes 84,000 to 66,000 for Obama -- 56% to 44%.

Then Obama gets 4.55 delegates in the second district, 1.05 in the first, and 2.8 in the state as a whole -- guess what! He gets 8.4 delegates, 56% of the total of fifteen. (Rounding those, Obama gets nine.)

Something similar is true for the state as a whole.

If anything, the district-based allocation helps Clinton relative to allocation based purely on the popular vote, because new voters tend to break for Obama (or at least they have in previous contents), and districts with a lot of new voters will be slightly undercounted.

Here in the Philadelphia area, we have an oddly-named chain of convenience stores named Wawa.

At Wawa, you can buy coffee for the following prices: 1ドル.09, 1ドル.19, 1ドル.29, 1ドル.39 for 12, 16, 20, 24 ounces respectively. This makes sense -- basically you pay $.79 for wandering around in their store taking up space and such, and then 10 cents for each four ounces of coffee.

However, things get weird if you bring your own cup (I'm talking about the "travel mug" sort here, not a paper cup). Then 12, 16, 20, 24 ounces cost 0ドル.85, 0ドル.95, 1ドル.05, or 1ドル.15 -- so far, so good. You save twenty-four cents by bringing your own cup.

32 ounces, in your own cup, is 1ドル.25. So now they're really starting to reward you for buying in bulk -- another ten cents gets you eight more ounces.

But then guess what happens? 64 ounces costs 2ドル.99. That,s right -- I can fill two 32-ounce cups for 2ドル.50, but filling one 64-ounce cup will cost 2ドル.99. If you extrapolate the linear trend from 12, 16, 20, and 24 ounces, 64 ounces should cost 2ドル.15. If I had a sixty-four-ounce travel mug, I'd go in there, fill it up, and try to get it filled for 2ドル.50 just to see how the cashiers explained it.

Perhaps they're trying to say that you really just shouldn't be drinking that much coffee. I'd have to agree -- and I'm a mathematician.

Another argument is that perhaps they are attempting to discourage people from taking that much coffee because then there's less coffee for the people after them, and people won't be happy if the store runs out of coffee. This may be true -- it seems a bit doubtful, though, since a typical Wawa store might have a dozen or so pots of coffee at once, each holding 64 ounces or so.