The Meaning of Probability

hamn damsters…looks like my topic didn’t go through; let’s try again…(good thing I kept a personal copy before sending)

I did a search of the forums and nothing turned up, so I suppose it’s fair game. Also, I’m posting this here rather than General Questions because (I think) it is more than a simple question with a simple answer; this may turn out to be wrong.

I believe the most common (layman’s, at least) definition of probability, the one I was taught in high school, is that probability is the “chance that a certain event will occur.” The emphasis here is on will, i.e., the notion of probability here is taken to be meaningful only when referring to future events.

I have trouble with this view. A few reasons: (1) its validity seems to depend on worldview: i.e., depending on your philosophy, you may think that there is no element of “chance” in the universe at all, e.g., in a mechanically deterministic universe. Nevertheless, even people who dismiss the concept of chance can (and many do, I don’t doubt) find the calculus of probability to be a very useful tool. (2) It is often useful to speak of probabilities even for past events (e.g., Baye’s Theorem); this is sometimes couched in the terms “confidence” and “likelihood.” Do these fundamentally mean the same thing as “probability”?

I prefer to think of probability as appealing not directly to chance, but instead to information that is deducible from known information, or expectations that can be derived from known information. The kicker here for me is that since different systems (say, people) may be working with different pieces of information, probabilites are inherently dependent on the system that is calculating the probabilities; probabilities are not absolute. The same event may have different probabilities for different observers. To one person, a particular event may be certain (or impossible), but to someone working with less information, the event may have a non-trivial probability. Also, by appealing to the concept of information and not “chance”, we can avoid hassling and discrimination between past and future events. Moreover, probability is meaningful regardless of whether the universe really functions “probabilistically” or not; whether or not God plays dice with the universe, as it were. This notion does however complicate matters, since probabilities for events are now only defined in the context of a set of pieces of information. In many cases, though, everyone might have exactly the same information.

You may have noticed that the last paragraph is rather vague. I haven’t ventured into more detail mainly because I would rather not reinvent any wheels and am giving you a chance to point it out before I waste my own time and the Straight Dope’s bandwidth. So I’ll open the floor now:

(1) Is there any current consensus among statisticians and mathematicians as to the meaning of probability? I know there are experts here (december, for one, I believe?).

(2) To what extent do you think the concept of probability is meaningful? (Future events only? All events? Not meaningful at all?)

The probability that a past event occurred, or that a present event is occuring, is either one or zero. But with less than perfect information, we are reduced to standard probability calculations. And that’s what it boils down to even if you believe in complete determinism–without all the information, you can only make guesses.

As for what exactly probability means…there is no consensus today. Many interpretations are in use, and there’s a case that all of them are right in the appropriate context.

I’m not going to claim to speak for all mathematicians and statisticians, so I’ll skip straight to your second question.

For me, the meaning of probability and other mathematical concepts originates in their descriptive power. Probability in particular describes the way certain experiments behave when they are repeated. For example, if I flip a coin 1000 times I should observe roughly 500 heads and roughly 500 tails. Probability is just a mathematical language to describe this observed behaviour of the universe.

I hesitate to assign any more “meaning” to probability than that. To me, it’s like asking what’s the meaning of the word “blue”. The only meaning of the word “blue” is this: there are all of these things that seem to be the same colour, and we find it useful to have a word for that. It’s an observed property of the universe. Probability is much the same.

The Weak Force, the book you want to read is, “The Foundations of Statistics” by L J Savage.

I would say that “probability” may be placed under the general heading “methods for representing data in summary form.”

Thus it is in the company of statistics (“3% of all elm trees died last year”) and qualitative ascription (“everyone in the room was happy”) and syllogistics (“all men are mortal, Socrates was a man, therefore Socrates was mortal”). In all these cases, a certain procedure is applied to a set of data, yeilding a definite result that summarizes the data.

(By “summarize,” I have in mind something like “represents the whole of the data in a compressed form without sacrifice of validity.”)

One thing to notice–often forgotten, to weird effect–is that “probability” is an inert and passive phenomenon, not the CAUSE of the data it represents. (Thus it is unlike most mechanical causal processes.) Except in metaphor, there’s no such thing as grabbing ahold of the probability that something will happen and reshaping that probability to force the thing to happen. The work must be done on the thing, not its description.

If you think no one would ever fall into that mental trap, try working with probabilities for a few years.

As the previous posts have already covered question #1 much better than I could ever manage, I’ll skip to #2.

I think that probability can be used to consider all events, assuming that principles that allow probabilities to be assigned can be derived. It’s a tool, and a very useful one, but not necessarily one that reflects a greater “truth” in nature.

Does probability have some deeper significance? I have no idea, and I doubt anyone else really knows.

There is in fact a precise mathematical definition of probability (if, as ultrafilter seems to suggest, there are other definitions then I’ve never seen them). If you’re not a mathematician you probably won’t like it much:

(Non mathematicians may wish to look up some basic definitions of set theory, or just skip the details entirely).

A sigma algebra, Y, on a set X is a subset of the powerset of X satisfying the following properties:

  1. 0 (the empty set) and X are elements of Y.
  2. If y is an element of Y, the complement of y is an element of Y.
  3. If T is a countable subset of Y then UT (the union of T) is an element of Y.

A measure on a sigma algebra Y is a function M from Y to [0,infinity] (the positive reals and infinity) satisfying:

  1. M(0)=0
  2. If A_n is a countable sequence of disjoint sets of Y then M(UA_n) = Sigma M(A_n)

A probability measure is a measure P that satisfies P(X)=1.

The elements of the sigma algebra are called events. For an event E, P(E) is called the probability of the event.

You might not agree that the definition of probability and the meaning of the probability are the same thing, in which case the above was mostly irrelevant to your question. However, there is a sense (which I happen to agree with) that the definition is the only real meaning you can give to something - the rest is just analogy. I suppose one way (the most simplistic way really) of thinking about it is that the events are things that ‘could happen’ and probabilities represent their relative potential for happening. Personally, I don’t like that much.

Essentially what I’m saying is this: Probability is a mathematical tool, nothing more. As such it frequently relates to the real world, often in surprising ways. Asking how it is useful in the real world is a good question. Asking what it is is also a good question. Asking what it ‘means’ as it is applied to the real world… not a terribly helpful (meaningful?) thing to ask IMO.

The question is not one of definition, but of interpretation. It’s the subjectivists versus the frequentists, with a few others thrown in as well. No one uses a different definition; they just disagree on what exactly P(E) = p means.

IANAE, but think of it this way.

There exists a large body of statistical machinery. One of the technical terms it uses is “probability”.

How one chooses to apply those tools is a separate matter. I’ll leave somebody else to describe what the frequentist or classical framework involves.

Subjectivist or Bayesian interpretations have something of the flavor as that given in the OP.

You might try searching with the phrase “Bayesian” or “Bayes”. Don’t forget that the default search scope is as of yesterday: you will have to reset that.

december - Thank you for the reference.

Math Geek - You may be right: that may be the best way of looking at probability, and in many ways it is the most useful (and is closely related to practical calculation of probability). I do have this nagging suspicion, however, that it is more of a useful heuristic than a real interpretation. Some “experiments” may not lend themselves to repetition at all, but I think even in some of these cases the concept of probability may be as meaningful as elsewhere.

kitarak - Thanks for the definition; I have never seen probability defined rigorously that way. However, as ultrafilter says, my main concern is the precise real-world interpretation of ‘P(E) = p’. Does it mean that it is impossible (assuming 0 < p < 1) to determine whether E will occur, but we can, in a sense, “expect” E to a certain degree indicated by p? Does it merely mean that events similar to E will occur on average 1000p times out of 1000? Or something else entirely? Maybe it is just an issue of context.

I have a book entitled Philosophical Theories of Probability by Donald Gillies. In it, he describes the major interpretations of probability that have come up in the last few centuries, and argues in favor of a contextual approach. You may want to check it out.

Well, in general the future is uncertain (quantum mechanics and all that), but probability theory is very useful for physics and prediction, as I’m sure you don’t need to be told.

Incidentally, it can be proven that an event E with probability p will on average occur a fraction p of the time.

You define two events A and B to be independent if P(A intersection B) = P(A) P(B).

Then if you have n independent ‘trials’ of an event E (basically a new event space where each event is a sequence of ‘yes/no’ n long, with the probability of each particular place being yes is P(E)) then the expected proportion which will be yes is p, and the probability that the proportion will be outside any interval surrounding p tends to 0 as n tends to infinity. (No, I’m not going to prove this here).

Expected values of course have a precise technical definition as well. Unfortunately I must confess I don’t remember exactly what it is in the most general case off the top of my head. Also it would require me to define random variables, etc. as well and you really don’t want me throwing more maths at you, do you? :slight_smile:

ultrafilter - that looks like what I need. I’ll try to get my hands on it as well as The Foundations of Statistics.

kitarak - Actually, I vaguely recall seeing a definition of this using inner product spaces…but the precise definition is really besides the point; as I said, what interests me is what it means in the physical universe.

Besides…isn’t mathematical precision bittersweet? You know, with Gödel’s Theorem and all :wink:

I’m suprized that nobody explicitly brought up the semantic observation between the usage of possibility and probability.
It’s possible to flip a perfectly ‘even’ coin forever and have it land on only one side; it is not probable.
It is possible that God created the universe, it is not probable.

I’m not exactly sure how that figures into the formulative definition of probability though. I’d be interested to hear a comment about it.



But lets not go into that here. Hijacks and all.

From a mathematical standpoint, possibility isn’t that big a deal. Events that are impossible have probability zero. Events that are not impossible may have any probability. Other than that, we don’t worry about it too much.