The title says it all. But its presence implies, and it is true, that people often mistake the two.
If you decide to bet fifty bucks the USA beats Germany, I cannot learn the probability which you formed for that proposition. That is, you had some list of premises probative of “USA beats Germany” (including tacit premises on the date of the game, knowledge of laws of the game, etc.). Now if I knew those premises, I with you would be able to deduce the probability of P = “USA bests Germany.”
Unless you’re a mathematical geek, that list of premises will be vague, in flux, imprecise, not articulable. The probability of P given this imprecision won’t be a number. Most probabilities aren’t numbers. If you say, “I think there’s a good chance P is true”, then this “good chance” is not a number.
And there’s no reason in the world it should be—though because of a strange desire that everything should be quantifiable, some insist all probabilities, and all decisions, must be numbers. This is yet another doorway over-certainty enters. Forcing somebody to state a unique single number for a probability when the evidence (premises) under consideration do not warrant it, and indeed say the probability is “fuzzy”, is always a mistake.
But even if you are a mathematical geek and have compiled a set of premises that allow a quantification of P—and there’s no a priori reason the mathematician’s premises and probability are superior to the sport bettor’s premises and probability—there’s no direct way to move from that P to a decision. Probabilities aren’t decisions. In fact, it is the conflation of the two, decision and probability, that lead some to conclude that all probabilities must be unique numbers.
Now there exists a field called decision analysis that shows how to take probabilities as inputs to a function which provides a decision. But there is a choice of function depending on how one views the consequences of making a decision. That’s a subject for another day. For us, we have to understand there is another input, which are the things risked and rewarded. For sports betting, these things are money, but for other situations they could be anything.
Money sure sounds like a number, and it is. But the difficulty is knowing how much money to risk and how much one would like as a reward. Figuring these kinds of numbers is just like figuring probabilities: they require precise, explicit premises which allow us to deduce the amounts. You won’t have these with betting whether P is true, just like you won’t have the exact premises which allow a deduction of the probability of P.
The same math geek can, of course, invent a list of premises which allow deduction of the amounts, but there’s no telling—in advance—whether his premises are superior to yours. Forcing a number to be precise when it doesn’t want to be allows you to make formal calculations, and this exercise gives the appearance of rigor and certainty. And that’s the problem: appearance.
This is the problem with decisions of major consequence. Take rampant, out-of-control global warming (which will strike any day now). Economists plug fixed probabilities of doom and the monetary values of that doom into gorgeous equations and out comes answers. These answers are the reified. They become the reality; rather, conditional prophecies of the form “Unless thou repent of carbon dioxide and put us in charge of all aspects of your life, you will pay exactly this much.”
Anyway, that’s beside the point—which is that probabilities aren’t decisions. Knowing the probability of anything, unless that probability be an extreme probability (0 or 1), does not tell you what to do.
I’ve decided to extend the special on typos through Saturday. Enjoy.