Probability is screwy, and we statisticians do a horrible, rotten job of teaching it. The first thing students learn in normal statistics classes is about “measures of central tendency” or some such thing. The idea of what probability means and why anybody would have the slightest interest in “central tendency” is never broached. As a consequence, students leave statistics classes with a bunch of half-remembered formula and no clear idea of what probability is.
This is unfortunate, because it allows educated men like Rolling Stone’s Bill McKibben to write the following:
June broke or tied 3,215 high-temperature records across the United States. That followed the warmest May on record for the Northern Hemisphere — the 327th consecutive month in which the temperature of the entire globe exceeded the 20th-century average, the odds of which occurring by simple chance were 3.7 x 10-99, a number considerably larger than the number of stars in the universe.[see note at bottom of page]
Poor man! Poor readers! McKibben actually believes he has said something of interest; he has worked himself into a lather over these numbers and goes on to say things like “the seriousness of our predicament”. McKibben figures that such a small number can only mean that we are doomed—unless, of course, massive amounts of money is taken from this country’s citizens and given to its politicians to apply as they see fit.
Now over the last week I tried to explain, via two examples, just what probability is and what it isn’t, and why numbers like McKibben’s aren’t of the slightest interest. See this post about global warming and this one about nine feet tall men. And if you find yourself disagreeing with me, read this one about foundations. You must at least read the first two posts because I assume it below.
What Probability Is
Suppose I let the symbol Q stand for “There are no men taller than nine feet,” and the expression D = “I observe a man 8.979 feet tall.” Let’s take this equation, or as some readers prefer to say, expression:
(1) Pr(D | Q)
and try to solve it.
Equation (1) is a matter of logic. It is just the same as Lewis Carroll’s French speaking cats: We know that if R = “All cats are creatures understanding French and some chickens are cats” that the proposition F = “Some chickens are creatures understanding French” is true; that is Pr(F | R) = 1. And this is so even if nobody ever, not ever never, in no possible world in no possible time, never never never measures or observes or sees or posits on genetical arguments any cats understanding French. It is true even if we learn tomorrow from God Himself that He has decreed that it is a logical and physical impossibility that any cat could understand French. F given R is true and that is that: and it is true because, again, logic only makes statements about the connections between propositions. Logic is mute on the propositions themselves.
All logic, which is to say all probability, because it is solely interested in the connection between expressions, must regard propositions as fixed. In any given equation, we cannot add or subtract from these expressions: we must leave them as they are: they are not to be touched: they are sacrosanct: they exist as they are and are carved out of uncuttable stone: we are forbidden upon pain of death to manipulate them in any way. For I testify unto every man that heareth the words of these theorems, If any man shall add unto these propositions, God shall add unto him the plagues that are written in Greenpeace press releases: And if any man shall take away from the words of these propositions, God shall take away his part out of the Book of Life. I am not sure how much more of a dire warning I can issue. Don’t touch Q or D!
Equation (1) says that assuming Q is true, assuming, that is, that there are no men taller than 9 feet, that it is true that there are no men taller than 9 feet, that it is impossible there are men taller than 9 feet, that God himself has willed that there are no men taller than 9 feet, that in any possible world there cannot be men taller than 9 feet, that it is just a fact, immovable, imperturbable, irrevocable that no man can be taller than 9 feet—even if we want one to be, even if we can imagine it to be so, even if real men are actually observed to be taller than 9 feet, even if you yourself are 9’1″—given, as I say, all that, what is the chance you see a man a quarter-inch short of 9 feet?
Well, on reading D to mean seeing a man shorter than 9 feet, (1) is certain, i.e. Pr(D|Q) = 1; or on reading D to mean seeing a man precisely 8.979 feet—the actual writing of D after all, and we know we should not touch D—the best we can say is 0 < Pr(D|Q) < 1 because we have no information on how heights are distributed; all we know is that heights are contingent, meaning it is not certain (given the information we have) that all men must be precisely 8.979 feet. And therefore all we can say is “I don’t know.”
We must judge equation (1) as written! Not as we imagine it to be written, or how it might be written differently is we change the meaning of Q and D. Or about how we feel about Q and D. How it is written and nothing else.
It’s kind of funny, but if we turn probability into math there wouldn’t be the slightest interest or confusion. Suppose instead Q = “X < 9” and D = “X = 8.979” where X is just some number unrelated to any physical real thing. Then Pr(D | Q) no longer seems mysterious. In this case it’s hard to see where to add bits about, “In my opinion, we might see X larger than 9” or “I would suspect that if X did equal 8.979 then X will be greater than 9.” Indeed, if anybody did announce the latter, you would regard him as eccentric. You’d say to him, “Listen, pal. These are just numbers. They don’t mean anything. And by assumption, no number can be greater than 9. So you are speaking out of your hat.”
Or change them again: Q = “Just half of all winged blue cats who understand French are taller than 9 feet” and D = “Observe a winged blue cat who understands French standing 8.979 feet”. Once again, we are not tempted to change Q and D and we interpret them as written.
Today’s lesson: don’t touch the propositions!
In Part II: McKibben’s Fantasy
If there were only 3.7 x 10-99 stars in the universe, there would not even be 1 star. 3.7 x 10-99 is of course less than 1.