Probability
What is it for something to be probable? This is a question that is probably all over the philosophy journals, but I don't have a very good idea of how to look up philosophy papers by topic. So let's reinvent the wheel, and investigate what probability is.
It's not an easy question. First, let's decide (1) whether we're talking about the probability of a single event occurring, or (2) the probability of, say, random event E occurring N times out of M. In the latter case, all we can say is that if the probability of event E occurring N times out of M is N, then event E is most likely to happen N times. This has some implications, of course; if you are in a gambling pool, bet on N. But what it does not do is reduce probability to anything. All we can say is that there is some probability, Q, of E happening N times (and some probability Q1 of E happening N-1 times...). We have changed the second problem for the first.
So let's look at the first problem. What is it for a single, one-time event to be probable? For instance, it's possible that an asteroid will hit the earth in 2012. (possibility is its own problem, but I'm satisfied there's a good explanation for what it is.) We universally agree that this asteroid impact is improbable, but what do we mean by that?
Perhaps we mean that the smart money is on the asteroid not hitting. That if (A) a million people bet against the asteroid hitting, and (B) a million people bet on its hitting, then in 2013, then (A) will be richer. But that's not necessarily true. If the asteroid doesn't hit, then (A) will make money, but if it does hit (and it is possible), then (B) will be richer. So all we can say is that (A) will probably come out ahead, but that doesn't help at all.
Do we want to say that there are more possible worlds in which the asteroid does not hit than in which it hits? That would be weird. First of all, is there a finite number of possible worlds? Doubtful. If not, of course, "60% of all possible worlds" doesn't mean anything. So assume there is. The trouble is, if physics is determinate (yes, quantum mechanics, but that has no effect on an asteroid in space) then in all possible worlds, the fact of the matter is that the asteroid will or won't hit the Earth. At best, there are a few possible worlds in which billions of rogue neutrinos knock the meteor off course, but the probability of that happening has nothing to do with our intuitive estimation of the probability of impact.
It seems more likely that the belief that the asteroid won't hit is based on some complete guesses. Something like this: "The asteroid won't strike the Earth unless there is a star at point S to deflect it towards us. We can't detect star S, because the moon is in the way, so we have to guess whether it exists. So, uh, say that the probability that star S exists is 1%. Done!"
Some astronomer is going to have to correct me on the details, but I'm sure all of our odds-making works kind of like this. So-and-so will happen unless X and Y and Z, or W and V happen. And that may be true, but what are the probabilities of X etc? We guess. Most likely, our guesses are based on the outcome of similar cases in the past -- if it was later discovered that 1% of postulated stars behind the moon exist, then run with that.
So we reduced the first question to the second. This is not progress. It's frustrating too, because probability is a real thing. The smart money is on the asteroid not hitting. And I say this as a man who hates astronomy.
It's not an easy question. First, let's decide (1) whether we're talking about the probability of a single event occurring, or (2) the probability of, say, random event E occurring N times out of M. In the latter case, all we can say is that if the probability of event E occurring N times out of M is N, then event E is most likely to happen N times. This has some implications, of course; if you are in a gambling pool, bet on N. But what it does not do is reduce probability to anything. All we can say is that there is some probability, Q, of E happening N times (and some probability Q1 of E happening N-1 times...). We have changed the second problem for the first.
So let's look at the first problem. What is it for a single, one-time event to be probable? For instance, it's possible that an asteroid will hit the earth in 2012. (possibility is its own problem, but I'm satisfied there's a good explanation for what it is.) We universally agree that this asteroid impact is improbable, but what do we mean by that?
Perhaps we mean that the smart money is on the asteroid not hitting. That if (A) a million people bet against the asteroid hitting, and (B) a million people bet on its hitting, then in 2013, then (A) will be richer. But that's not necessarily true. If the asteroid doesn't hit, then (A) will make money, but if it does hit (and it is possible), then (B) will be richer. So all we can say is that (A) will probably come out ahead, but that doesn't help at all.
Do we want to say that there are more possible worlds in which the asteroid does not hit than in which it hits? That would be weird. First of all, is there a finite number of possible worlds? Doubtful. If not, of course, "60% of all possible worlds" doesn't mean anything. So assume there is. The trouble is, if physics is determinate (yes, quantum mechanics, but that has no effect on an asteroid in space) then in all possible worlds, the fact of the matter is that the asteroid will or won't hit the Earth. At best, there are a few possible worlds in which billions of rogue neutrinos knock the meteor off course, but the probability of that happening has nothing to do with our intuitive estimation of the probability of impact.
It seems more likely that the belief that the asteroid won't hit is based on some complete guesses. Something like this: "The asteroid won't strike the Earth unless there is a star at point S to deflect it towards us. We can't detect star S, because the moon is in the way, so we have to guess whether it exists. So, uh, say that the probability that star S exists is 1%. Done!"
Some astronomer is going to have to correct me on the details, but I'm sure all of our odds-making works kind of like this. So-and-so will happen unless X and Y and Z, or W and V happen. And that may be true, but what are the probabilities of X etc? We guess. Most likely, our guesses are based on the outcome of similar cases in the past -- if it was later discovered that 1% of postulated stars behind the moon exist, then run with that.
So we reduced the first question to the second. This is not progress. It's frustrating too, because probability is a real thing. The smart money is on the asteroid not hitting. And I say this as a man who hates astronomy.
posted by apk01004 at 2:31 PM
2 Comments:
thank the good lord for rogue neutrinos!
Van Fraassen is a philosopher of probability who discusses betting as an epistemic measure of our own belief probabilities. For a discussion of possible worlds, the most famous is David Lewis. But I don't think he uses them in quite the sense you do. First of all, there are no finite possible worlds, they are infinite, and we don't use them to determine probability, but more to determine counterfactual questions like "okay, X didn't happen, but what if it did."
Cool, someone discussing philosophy. Don't know if you were actually curious to read more on probability, or if this was a one-day musing, but there's my limited knowledge on the subject.
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