Monday, April 14, 2014

Billion Dollar Brackets and Fine Tuning

Backgammon PrecisionDice

For this year's NCAA Men's Basketball Tournament, Warren Buffet offered to pay $1 billion to anyone who could correctly predict the outcome of every game.  Unsurprisingly, no one won.  Many stories and commentaries have cited numbers about just how improbable it is to pick every game correctly.  What none of these stories and commentaries seem to have noticed is that the question is meaningless.

The usual comparison is with a lottery.  Lotteries are run so that every possible combination of numbers is equally likely, which means that no one strategy is any better than another.  Choosing "1 2 3 4 5" for Powerball works just as well as "2 3 5 7 11" (the first 5 primes) or "8 15 16 23 42" (from the Lost sequence) or any other sequence of 5 natural numbers less than 60.  (Don't play any of those sequences, though.  Because there is a "reason" for each of them, several other people are probably already playing them, which means that even if you won the prize, it is much more likely that you'd have to split the winnings!)

Most estimates of the "odds" of getting all the games right in a bracket challenge probably assume that the odds getting each individual game right are 50/50.  Since there are 32+16+8+4+2+1 = 63 games in a bracket (not counting the play-in games), this would make the odds of a random bracket winning 1 in 263, or 1 in 9.223 x 1018.  However, no 16 seed has ever beaten a 1 seed in the NCAA basketball tournament, but 93.75% of brackets chosen completely at random will have at least one 16 seed beat a 1 seed.  Clearly, there are better strategies for filling out a bracket than just flipping a coin.

So what is the best strategy for filling out a bracket?  No one knows!  My strategy is to choose the winner of each matchup so that the odds of the team with the lower seed winning are (15 + difference in seed numbers)/30; that way, the odds of a 1 seed beating a 16 seed are 100% but the odds of one 1 seed beating another 1 seed (in the Final Four or Championship Game) would be 50%.  This assumes that the people on the committee that sets the seeding really know what they are doing, which is debatable -- but what isn't debatable is that they know more about college basketball than I do.

Of course, there is a lot more information available -- the outcomes of games through the season and the order in which they came, the stats for individual players, the general strategies of the coaches, and the announced injury situations.  No one really knows how to put all this together!  Then there is information which is not publicly available (exactly how the players feel, physically and emotionally) and there are unpredictable things that will happen over the course of the tournament.

In the case of Powerball, any strategy is the ideal strategy, but in the case of a basketbal bracket, the ideal strategy is unknown.  If the strategy is unknown, how can we know how likely it is to win?

At any rate, let me pretend that someone who really knows basketball can choose the 1 vs 16 matchups correctly 100% of the time and the other ones 70% of the time.  His odds of getting all 63 games right are 0.759, or 1 in 1.378 x 109.  Probably no one is really that good, and he would still face long odds, but he is a billion times more likely to fill out his bracket correctly than the crude estimate had indicated!

Let's go a little further.  Suppose someone is a super-genius at college basketball and can pick games correctly 99% of the time (and I'll give him the 1 vs 16 matchups for free).  Someone like that would already have his own billion dollars, but he's going to play anyhow.  His odds of filling out the whole bracket correctly are 0.9959, or 55.27%.  To me, that is the best measure of just how hard this is!

Unfortunately, it's not hard to find people quoting probabilities as though they were meaningful when they really are not.  What are the odds that your marriage will end in divorce?  Sorry, national stats are really useless in this case, because you are not a typical person.  No one is.  Your background and personality (and that of your spouse) are unique, and they have everything to do with whether you two will decide to remain together.

One particularly frustrating (to me) use of bad statistics comes in the fine-tuning argument for a Creator.  This is based on the fact that there are 25 free parameters in the Standard Model; these are physical constants with values that can only be determined by experiments, not on the basis of some principle we know.  In several cases, a change of a few percent to one of these parameters might have caused the Big Bang to collapse on itself again, or for stellar nucleosynthesis to create a world in which no atoms are more complex than hydrogen or in which all stars are neutron stars.  In such a universe life as we know it would be impossible.  What are the odds that this happened by chance?

Once again, we do not have enough information to answer the question, and it is dishonest to pretend we do.  Since these are "free parameters", we obviously need additional information to determine their values -- but we also need additional information to know anything about the distribution from which they might be randomly chosen.

Dice (typical role playing game dice)  

Think of it this way:  Maybe the universe is the way it is because, Einstein notwithstanding, God did throw dice to determine the answer to the Ultimate Question of Life, the Universe, and Everything, and 42 was the number that came up.  We can further specify that all the dice were fair and identical, and that the most likely value for the throw was 63.  What are the odds that 42 came up?  The number will depend on whether it was 6 20-sided dice (6 x 10.5 = 63) or 18 6-sided dice (18 x 3.5 = 63).  Even in this simple example, if we do not know what kind of dice were used, we cannot answer the quesion.

Saturday, April 12, 2014

Marriage and Materialism

It has recently occurred to me that one of the main reasons that "gay marriage" has garnered so much public support is the widespread acceptance of materialism.  Before I go any farther with this, though, let me get a few things out of the way.
  • When I say materialism, I do not mean consumerism.
  • I find materialism entirely unconvincing, in part because only a spirit can be convinced or deceived.  To claim that we are bodies deceived into thinking that we have minds has always struck me as the most complete nonsense; it would make more sense to think we are souls deceived into thinking we have bodies, though I don't agree with that, either.  The point of this post is not the strengths and weaknesses of materialism, though, so I will leave it at that.
  • Acceptance of "gay marriage" is not determined by whether or not materialism is embraced.  The Soviets were materialists, but they remained basically sexually conservative.  Many Protestant clergy accept "gay marriage", but they presumably reject materialism. 
Anim engrenages droits

A good argument could be made that Americans (and Europeans) are, if anything, more superstitious now than at any time in the recent past, and that might seem to be at odds with them increasingly embracing materialism.  Not really, at least in this case.

Take, for instance, the fascination with "paranormal investigations". Even though they use the word "spirit", they have a very corporeal idea of what a spirit is:  to "explain" survival of the soul after death, they use conservation of energy; they look for spirits using electromagnetic field detectors; they try to explain why some places are haunted in terms of their vague understanding of the electrical properties of quartz and water.  They try to explain ghosts as phenomena of natural science, even though their "science" is merely pseudoscience.

Likewise, zombies and vampires are wildly popular, but they are almost always shown as people who didn't really die (though it may have looked like they did), but rather sufferers of some viral infection.  Science fiction/fantasy is not much better -- that genre consistently misreads "Any sufficiently advanced technology is indistinguishable from magic" to mean "Anything we can imagine magic might do can be done by a sufficiently advanced technology," and they proceed to make a weird mishmash of superstition and pseudoscience that fails to make any sense.  For example, in the Star Trek universe all humans are apparently atheists -- at least that was the story while Gene Roddenberry was still alive -- but (to name a few) the "prophets" or "wormhole aliens" from Deep Space Nine were functionally gods, as were the members of the Q Continuum.  At least they were more powerful than most or all of the Greek and Norse gods.  These beings may involve exotic materials, maybe materials that involve other dimensions (a la string theory), but it is always stated or assumed that material explanations (of some sort) are entirely sufficient.

In contrast, consider the objects of mathematics:  numbers, geometric shapes, etc.  Although we may have 1 apple, 2 apples, 3 apples, or whatever, certainly the numbers 1, 2, 3, ... do not exist physically, yet it is hard to shake the feeling that they somehow have an existence independent of us and that we discover them rather than invent them.  Not everyone agrees on this point; some think that mathematics does not describe a kind of non-physical reality, but that it only tells us something about how the way the human mind works.  To me, at least, that is very unconvincing; surely a civilization of intelligent aliens would know about the natural numbers (and all the integers, and the rational numbers, and the real numbers, ...) and Euclidean geometry (and hyperbolic geometry, and spherical geometry, ...).  The aliens would know that the ratio of the circumference of a Euclidean circle to its diameter is a constant, and that the value of that constant is an irrational number that is approximately 3.14159265.

It should be pointed out that just because we know the integers and the operations of addition, subtraction, multiplication, and division does not mean that we know everything there is to know about them, nor does it mean that we cannot make wrong guesses about them.  Gödel's incompleteness theorems show that there are truths about the integers which we cannot prove axiomatically.  It is not yet known whether or not the Twin Primes Conjecture is true, though significant progress has been made on that question recently.  The Polya Conjecture has been shown to be wrong, as has the Mertens Conjecture.  The key thing is that if a conjecture can be proved to be true, like Fermat's Last Theorem was, then it is true everywhere, at every time, and for everyone; if it is proved to be false, it is false everywhere, at every time, and for everyone.

Those who claim that the Natural Law exists (I am one of them) insist that the Natural Law is something similar:  like mathematics, it exists independently of human opinion or knowledge.  In some ways, medicine might be an even closer analogy, because medicine is about humans (as the Natural Law is), is more contentious, and rarely if ever has the full rigor of mathematics (what does?).  Yet again, truth is not just a matter of opinion.  Even when it was believed that tobacco could cure cancer, it was still a health hazard.

It is silly to insist that serious materialists would be unable to use adjectives or verbs, but materialist philosophy taken seriously will impact how language is understood -- particularly the subjunctive mood.  "People should not smoke tobacco" would have to mean something like "Smoking tobacco leads to health problems, which most people and societies wish to avoid."  Even that formulation is a bit iffy, since "smoking tobacco" is an abstraction, and "people" and "societies" are universals.  A materialist might still be willing to use them (it is hardly possible not to), but he would consider the universals to be merely nominal -- just a name we impose for convenience, not a reality in itself -- that comes in handy for fuzzy thinking.

I think no reasonable person could insist that this is always wrong.  For example, it has become clear that there is no sharp distinction between a comet and an asteroid, or between an asteroid and a planet, or between a planet and a brown dwarf, or between a brown dwarf and a star.  It is very useful to have such words to narrow down what we are talking about, but no matter how we define these categories, there will be objects (not necessarily in our own solar system) that straddle the boundaries.  These words are names that we impose on nature, not ideas that we discover in nature.

Likewise, it would not be reasonable to insist that there is a universal ideal of American football that is discovered.  At any given time, there are several different sets of rules for American football -- at the high school level, the college level, and the NFL, for example -- and the details of the rules change from year to year.  It is probably a safe bet that no other society within the observable universe plays football with exactly the same rules set by the NCAA for the 2013 season.

My contention is that a materialist worldview has been widely absorbed by the public.  This would explain why some people clearly have so much trouble understanding that there are some parts of reality the government cannot change by passing a law or issuing a ruling from the bench.  The government can pass a law making it illegal to smoke tobacco, but if the government passes a law that the smoking of tobacco cures cancer, that law will have no effect.  In 1897, a bill was proposed in Indiana to establish by legislation based on a claim by Edwin Goodwin that he had discovered a way to square the circle (a known impossibility); the bill would also have had the effect of establishing one (or more!) different values of π from the one established by mathematics.  The bill did not pass, but even if it had, it would have changed nothing; as it is, it only made Indiana the butt of jokes.

The real disagreement, then, is over whether marriage is something with a fixed substance, or not; is it something like football, where we can change the rules as we please, or is it something like math, where we can't?   If many people today believe that all universals are merely nominal and that all laws -- of math and physics as well as the Congress -- express only the culture currently in power, it will be nearly impossible for them to understand, let alone persuade, supporters of traditional marriage -- and vice versa.

Every Child a Wanted Child?

Someone in Pakistan thought that was a good idea.

Monday, April 7, 2014

Proverbs 26: the Internet Chapter

Seriously, just read through it and think about what you have read in comments on blogs and news articles recently.  "Answer not a fool according to his folly, lest thou be made like him."  That's one for me; I am too often "made like him."  "As he that taketh a dog by the ears, so is he that passeth by in anger, and meddleth with another man's quarrel." We call these "trolls" today.