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?)