Statistics vs. the Lottery

So apparently there's an article in Harper's (can't see the original; link below is commentary at Forbes) on the following subject -- Joan R. Ginther has been "outed" as a statistics professor with a PhD from Stanford, who possibly deduced the winning-ticket lottery distribution schedule in Texas, and has hit multi-million dollar jackpots 4 times in the last decade. Notes:

(1) While I don't see any assertion of any way in which this would be illegal, the overall tone is clearly one of how-dare-she-think-she-can-get-away-with-this. “When something this unlikely happens in a casino, you arrest ‘em first and ask questions later,” says a professor at the Institute for the Study of Gambling & Commercial Gaming at the University of Nevada, Reno.

(2) "The residents of Bishop, Texas seem to believe God was behind it all."

http://blogs.forbes.com/kiriblakeley/2011/07/21/meet-the-luckiest-woman-in-the-world/

Less Time to Learn

Hypothesis: The less time students have to learn, the higher their testing scores are.

This has been a suspicion of mine for a while now. For example, I find that my accelerated summer/winter modules (6-week courses) generally outperform my normal fall/spring modules (12-week courses) in the subject material, testing procedures, etc. I'm guessing that the major factors involved are (a) a greater focus and more connections with the given subject material, (b) fewer competing courses being taken at the same time, vying for mental attention, and (c) simply less time and opportunity to forget stuff from class to class, which I feel is a real issue for many of my students. (Countering factor might be: Maybe more dedicated students register for summer/winter courses?)

So this summer I had an excellent accidental experiment in this regard. I'm teaching two statistics classes in parallel on Mon/Wed and Tue/Thu nights. There was a weird burp in the schedule (specifically, the Mon Jul-4 holiday) that caused one class to be ahead of the other by one evening's lecture. So heading into the last test (partly on hypothesis tests and P-values), the Mon/Wed class was first introduced to the subject just 2 weekdays (48 hours) in advance of the test, whereas the Tue/Thu class had a whole week (7 days) to see P-values and study for the test (including, obviously, a whole weekend).

So I was rather concerned that the Mon/Wed class was being unfairly put upon, what with such a short window in which to study, and on Wednesday they did seem to struggle. But then to my surprise it turned out that the Tue/Thu class found what was basically the same test even more challenging, and got a significantly lower average score on the same assessment.

On Tau

So recently there were some popular news articles with titles like, "Mathematicians Want to Say Goodbye to Pi" -- first I've heard of it, and of course initially it sounded ridiculous (I guess that's the point of news-article title-writing, eh?) The gist of it is that in theory, when dealing with circles, it would easier to exchange the value pi = circumference/diameter for tau = circumference/radius, i.e., tau = 2*pi.

And actually, that very quickly hit me as something that would be very nice to have. It would make a lot of trigonometry and calculus easier. The number of radians in a circle would simply be tau (instead of 2*pi). Perhaps most important for me, circles are inherently defined by their radius (all points a given distance from the center), not by their diameter.

Now my first attempt at an objection was the formula for a circle's area, which would get ever-so slightly more complicated, switching from A = pi*r^2 to A = tau*r^2/2. But that's a small thing, and in fact it reminds you of the fundamental integral(r)=r^2/2 which is used to derive it in calculus (instead of a disappearing denominator trick, canceled by the constant 2*pi).

The other thing that just occurred to me -- and motivated this post -- is what it does to Euler's identity, e^(i*pi) = -1 (or however you want to move the terms around). Now, I may be an angry crank, but if I think deeply about this celebrated identity (it was voted "most beautiful formula" in the Mathematical Intelligencer, 1990; a post which I have taped on the wall over my computer), it's not terribly interesting; granted that the imaginary part of the exponential function is a rotation in the complex plane, and coincidentally pi happens to be half a circle, i.e., landing on the point (-1, 0). If we used tau more commonly, then the triviality would be more apparent: e^(i*tau) = 0, and no one would get as worked up about it anymore. Or maybe people would think it's even more "beautiful" then, hell, I don't know. :-)

Am I going to try to switch the thousands-year legacy of using pi to tau? Not me, man, I've got enough to do without quixotic crusades. But yeah, if I could pick different historical legacies the options for (1) switch pi to tau, and (2) switch electrical current signs (link), would be near the top of the list.

What do you think?

Edit: Of course, e^(i*tau) = 1 (not 0). [Knocks self on head.] Maybe that actually is more beautiful.

Math in the Internet Age

From Mike Jones, commenting on the proposed proof to the Collatz Conjecture:

The paper runs to 32 pages and we will have to wait for it to be checked for errors. Such is mathematics in the Internet age - no longer are proofs brought down from the mountain top in their perfection but they are thrown to the crowd to survive being torn apart.

Lindley's Paradox

The Wikipedia description of Lindley's Paradox asserts an example of opposite hypothesis-testing results between the Frequentist approach and the Bayesian approach.

The example is one of testing a certain town for the ratio of boy-to-girl births. The thing that violently strikes me here is the choice of the Bayesian prior: P(theta = 0.5) = 0.5, i.e., the advance assumption that it's 50% likely for the ratio to be equal to 0.5 (the other 50% chance spread uniformly between all points from 0 to 1).

I mean: What? Why would I conceivably assume that? If I broadly picture real numbers as being continuous, then my instinct would be to assume that it's almost impossible for any given number to be exactly the parameter value, i.e., I'd assume P(theta = 0.5) = 0. Even if I didn't reason that way, I otherwise have copious evidence that human births aren't really 50/50, there's very clearly more boys born than girls -- so if anything I'd choose that as the most likely prior value.

Is that really how Bayesians are supposed to choose their prior? (It seems atrocious!) Or is this just a fantastically mangled example at Wikipedia?

The Amazing Lottery

Stats observation of the day -- After every lottery you can say, "That number had only 1 chance in 175 million of coming up!". But, there's a 100% chance you can say that, every time it's run. It's only interesting or significant if you can predict the result in advance. Otherwise you have a fallacy called "data dredging": http://en.wikipedia.org/wiki/Data_dredging

See also:

• Testing hypotheses suggested by the data
  
• Multiple comparisons fallacy
• Prosecutor's fallacy
• Ecological fallacy

Other probabilistic fallacies.

Frequentism and LLN

I would be seriously keen to find this out: What difference is there between the frequentist interpretation of probability and simply a restatement of the Law of Large Numbers? Because I kind of can't see any. And why is the LLN never brought into any such discussion of probability interpretations?