Monday, January 16, 2006

Odds 'R

The second in the series of posts that stem from comments that weren't there the last time I looked comes courtesy of Dan, who pointed me towards Odds'R, which is a book of trivia presented in probabilistic form. e.g.:
If you had been arrested for witchcraft during the infamous Salem witch hunt and trials of 1692, what would have been your odds of being executed?
a) 1 in 4
b) 1 in 8
c) Zero--they all got away on their brooms.
Also there's a board game, and I'm sure any moment now, you're going to be able to buy the Official Odds'R Bow Tie and Mustache Set™ in a store near you (dude, that tie totally rocks).

Dan asks:
Is asking the odds on such specific questions as above still a problem within your argument that the "odds" are statistically meaningless because the question was asked within a framework where "meaning" does not correspond to numbers, thus the "odds don’t have the meaning that they appear to have?"
I, being a crass opportunist, will use this question to launch a larger discussion of what probability actually "means."

What can we actually infer from the above question about witchcraft (b is the answer they give as being correct--I know, I can't believe it wasn't c, that pinacle of well-timed comedy, either. Oh my god...witches, brooms...stop. I'm gonna pee)? Only that there were some number of people arrested for practicing witchcraft in Salem in 1692, say 24, and that 3 of them were executed. To infer what the question actually states as being true ("if YOU, the person now reading this sentence, were a person living in Salem in 1692 and you were arrested for practicing witchcraft, 12.5% of the time you would be executed") would be completely fallacious, meaningless, and frankly kind of a weird thing to infer. I can say with certainty that this did not happen to you. The odds of it are zero.

The phenomenon of Odds-Are-Oneness tries to make some sort of statement about what we tend to think of as fate--weird things happen to us, things that seem like they were spectacularly unlikely to happen, but did. But we also notice the "weird" things, and ignore the incidents of similar phenomenon that aren't "weird" to us. To review my very first example, nobody took note of the thousands of times that the New York City Pick Three Lottery came up, e.g., 1-6-5 on May 13th, 1995. Except, of course, to the people who won the lottery that day, at which point it suddenly seemed a lot like fate to them, and they, if they're like the majority of people in the world, probably went back and created some narrative about what those numbers "meant" and "why" they won.

But I also think the answer to Dan's question is unequivocally yes. These odds don't have anything to do with probability. They can't possibly apply to you imagining yourself into the shoes of an accused witch in Salem in 1692. Or in the case of another question found in Odds'R on a favorite topic of mine, "What are the odds that a US state has a law on the books challenging the validity of evolution?" (given answer, 17 in 50)--I'm not saying there's no meaning in that statement, but it sure isn't about probability. It's for sure not telling us that when the next territory or protectorate votes for statehood that there's a 34% chance someone will toss in an amendment saying, "We vote for statehood, and P.S., Puerto Rico also declares that Darwin did crack."

So the open question is, is there actually any such thing as logical discrete statistical inference? Can we logically infer from the fact that one out of every million plane flights ends up crashing whether or not we should get on the next plane? Does knowing that one out of every fifty persons who takes Vioxx experiences heart failure within two years tell us anything about whether we should take it? Don't get me wrong, probability and statistics are not meaningless. Casinos make money. Carbon 14 decays. A relatively predictable number of people will buy things from Amazon in the next year. Over a large sample size, statistics and probability are inexorable. The longer you flip a fair coin, the closer it runs to 50% heads, 50% tails. And you still don't know a goddamn thing about what the next flip is going to be.


Sam said...

OK, as you know from past comments and posts of my own, I'm totally with you on the OAO thing. And thus, 'chances of me being a witch in salem?' - meaningless. And 'odds of a state having a law challenging evolution' - just stupid, since they aren't really getting at the odds of anything, they are just saying that a certain number of states has these laws. All fine and dandy.

But then, at the end, you get really radical on us: "The longer you flip a fair coin, the closer it runs to 50% heads, 50% tails. And you still don't know a goddamn thing about what the next flip is going to be." Whoa! Don't I know that the odds are .5 that the next flip will be heads? If I were to say, I'll bet you $10 that out of the next ten flips 7 will be heads, wouldn't you say 'I'll take that bet'? And thus, looking forward into the future, with enough information known ISN'T there such a thing as logical discrete statistical analysis? It sounds like you're saying there's not, and that kind of blows my mind a bit. So I'd like a bit more explanation.

Transient Gadfly said...

Absolutely I'll take that bet. I mean, my god, the odds of flipping exactly 7 heads out of ten coin flips is...(does the math)...10!/(7! * 3!) in 2^10 ...(decides not to do any more math)...but it's pretty good odds for me. The thing is, there's a possible universe out there in which you'd win and I'd lose. If we ended up in that universe after ten coin flips, was my little logical inference wrong? Most reasonable people would say it wasn't, even though I lost.

Anyway, you've caught me. I am proposing something pretty radical. It's going to be another one of those cases where I won't disagree with the conclusion (I should take Sam's bet, duh. Besides, it's only ten bucks. Now, if we were betting my fabulous new car stereo, I might hesitate), but I'll end up proposing a new model. Our understanding of the implications of statistical inference has some problems. Alive/dead cats and a near-infinite number of possible universes splitting off from ours at every instant come to mind. Anyway, more explanation to come....

Transient Gadfly said...

can i offer tom stoppard on this?

R&G Are Dead starts, you'll recall, with ninety coin tosses that all come up heads. guildenstern lists the following possible explanations:
1) he doesn't know it but he's willing it to happen
2) time has stopped, and it's really the same experience, the same moment, over and over
3) god
4) well, it's always a 50-50 chance, so you can't be at all surprised that it comes up heads any individual time

1 seems ridiculous, but is largely the implication suggested by the notion "the odds are one," not quite in spirit, of course, but in fact. 2 seems on the face of it absurd but is in fact much the point of the dead/alive cat, einstein, and nobel prize winning physicists the world over. 4 is both TRUE and WRONG and many would, of course, swear one or the other about 3. in the play, the answer is (hurrah!) narrative -- the fate of these characters has long been set, not by destiny, not by chance, not by will, not by science, and not by god, but (close) by shakespeare. that is, this story has already been told, read, lived; it is always then different -- it's (a) play -- but also always the same. that is, just as there are no odds re: the salem witch trials as they have already happened and the data we'd analyze to predict is actually history, the same is true of the sort of things we think lend themselves to prediction now. we, our events, are not yet history, but any data we have to analyze already is.

Sam said...

Yep, OK, I'm delighted to have pushed you (both of you) further into your radicalism. Yay radicalism. By the way, it wasn't until the signature that I was at all sure who wrote the second comment. Yes, it was all about narrative, but it was also all about odds and Paul's posts are now all about narrative too, so....

Transient Gadfly said...

actually, we're the same person now. we got married. hello.

(and actually, seriously, it is interesting that when i read your blog, the first thing i do is scroll down to see which of you has written it.)

plus p. uses capital letters. weird.

sageblue said...

see, now, when are you going to start doing something useful for me with your this guy?

Anonymous said...

To the confused probalists: try flipping a coin 100 to 200 times and count the number of times it lands on heads 5 to 7 time in-a-row. Name your bet that on any 10 flip sequence there will be an even split; and compare the number of times there's an alternating heads to tails or vice-versa over 2000 flips.

Sam must have missed the point of "Survival of the witless"--and the fact that the book wasn't just about the literal probabilities of a given situation, but the humorous and counterintuitive outcomes of various events.