Probability

Mathematical Aspects of Probability

The mathematical foundations of probability are simple. When analyzing some scenario probabilistically, we select a space of basic possibilities, requiring that they be exhaustive (at least one must obtain) and exclusive (no more than one can obtain)—for example, in a toss of a pair of dice, the 36 possible ways the dice can land are usually taken to be the basic possibilities. Each possibility gets assigned a number in the range (0, 1), which represents its probability—1/36 in our example, assuming that the dice are fair and independent. A proposition (what mathematicians call an “event”) is something that turns out true according to some of the basic possibilities and false according to the rest (e.g., “at least one die lands 6”). The probability of any proposition is then simply the sum of the probabilities of those possibilities that make it true.

Standard axioms for probability theory fall out immediately: Given any propositions A and B,

• i.
  0 ≤ Prob(A) ≤ 1;
• ii.
  if A is a contradiction (true according to no possibilities), then Prob(A) = 0;
• iii.
  if A is a tautology (true according to all possibilities), then Prob(A) = 1; and
• iv.
  if A and B are incompatible (no possibility makes both true), then Prob(A or B) = Prob(A) + Prob(B). (Some presentations use an infinitary version of this axiom)

It is then usual to define the conditional probability of A, given B, as

• Prob(A, given B) = df Prob(A and B)/Prob(B), when Prob(B) ≠ 0.
• A and B are independent iff Prob(A and B) = Prob(A) • Prob(B). If so, then Prob(A, given B) = Prob(A) when the left-hand side is defined.

This presentation simplifies one detail, since we have assumed that the space of possible worlds is at most countably infinite; if it is uncountably infinite, then, typically, each basic possibility gets a probability of zero, and we apply techniques from calculus to integrate over sets of basic possibilities in order to arrive at nonzero probabilities for propositions. The axioms listed above still hold.

Beyond laying down the requirement that the basic possibilities be exclusive and exhaustive, the mathematical structure of probability does not dictate the choice of what they are, let alone what their probabilities are. In practice, some judgment is required in making this choice, so that subsequent analysis can proceed in the clearest and most illuminating fashion.

Metaphysical Aspects of Probability

Turn now to metaphysics: What is a statement of probability about ? For example, if a meteorologist [Page 762]asserts that the probability that it will rain tomorrow is 30%, what must our world be like for that assertion to be true ? Approaches to this question display a striking disparity.

On the classical approach—now fallen into disfavor—the probability of a proposition is simply the ratio of those basic possibilities according to which it is true to the total number of basic possibilities. This approach assumes, implausibly, that in any given application, there is a uniquely best way to select the set of basic possibilities and that each possibility in the set should be treated as equally probable.

...