Objective Probabilities

The common distinction between probabilities that can be based on frequencies known to hold in a sequence of repeatable events, and probabilities that concern unique events, and that therefore must be based on subjective opinion, is argued to be misguided. All events are in some relevant sense "unique", and, more importantly, all events can in a relevant sense be placed in classes of similar events. A formal calculus is described for accomplishing this in relatively simple but useful cases. * Almost everyone will agree that when our background in statistical knowledge is extensive enough, and when the case with which we are concerned is a "repeatable" event, then objective probabilities are appropriate, and these are the probabilities that should enter into the computation of expectation and into our decision theory. A great many people will also agree that there is another whole class of cases, in which we are concerned with unique events, in which we lack statistical knowledge, and for which we must turn to subjective probability or one of its surrogates. I propose here to argue against this distinction. Of course it is easy enough to argue this way in a purely philosophical vein: every event must be unique-it has its own spatio-temporal locus; and every event must belong to some class of events about which, in principle, we could have statistical knowledge. But this is not my point. My point is that from a down-to-earth practical point of view, from the point of view that seeks to compute probabilities and expectations for making decisions, the distinction between 'repeatable' and 'unique* events is not only untenable, but seriously misleading. B. Kinds of Cases Let us consider some examples of these alleged distinctions. Consider the toss of a coin. There is a classical 'repeatable' event: not only can we toss a coin over and over again; coins have been tossed over and over again, and in the experience of each of Rochester of us there is a large database of results of coin tosses (or an impressionistic resume of such a database). And we have physical grounds (i.e., grounds stemming from the laws of physics for thinking that coins land heads about half the time. And so we can regard that toss of that coin as a member of a class of tosses, of which we have reason to believe that half yield heads. (Alternatively, we might regard that …