A Primer on Probability

Probabilities can be associated with events or with propositions. (Remember that a proposition is a claim that can be true or false. More precisely, a proposition is defined as the meaning of a declarative sentence. Two sentences that have the same meaning are said to express the same proposition.) The probability of an event indicates how likely it is that the event will occur. The probability of a proposition indicates how likely it is that the proposition is true.

The notation ‘pr(X)’ represents the probability that proposition X is true. (Some authors use other notations like ‘p(X)’, ‘P(X)’, or ‘prob(X)’ instead.) The probability of an event can be represented with the very same notation, since the probability of an event can be identified with the probability that a corresponding proposition is true—namely, the proposition that the event will occur. For instance, if L is the proposition that you will win the lottery, then pr(L) is the probability that you will win the lottery, which is the same as the probability that L is true.

The notation ‘pr(X|Y)’ represents the conditional probability of X given Y—that is, the probability that X is true given the assumption that Y is also true. For instance, suppose Q is the proposition that a certain playing card drawn at random from a standard deck is the Queen of Hearts, and F is the proposition that it is a face card. Since 12 of the 52 cards in the deck are face cards, the conditional probability pr(Q|F) is 1/12, while the unconditional probability pr(Q) is 1/52.

Mathematically, probabilities are numbers from 0 to 1 that obey certain rules, including the following:

  1. The probability of a tautology (a logically necessary truth) must be one.
  2. The probability of a contradiction (a logically necessary falsity) must be zero.
  3. The probability of a disjunction must be greater than or equal to the probability of each disjunct: 

    pr(X ∨ Y) ≥ pr(X)    and    pr(X ∨ Y) ≥ pr(Y)

  4. The probability of a conjunction must be less than or equal to the probability of each conjunct:

    pr(X • Y) ≤ pr(X)    and    pr(X • Y) ≤ pr(Y)

  5. When propositions are mutually exclusive and exhaustive, the sum of their probabilities must be exactly one.

Propositions are mutually exclusive when no two of them can be true at the same time, and they are exhaustive when there is no way for them all to be false at the same time (in other words, there are no further possibilities). For example, the four propositions (X ∨ Y), (~X ∨ Y), (X ∨ ~Y), and (~X ∨ ~Y) are exhaustive, since there are no further possibilities: at least one of them must be true. But they are not mutually exclusive, since some pairs of them can be true at the same time: for instance, the first two are both true when Y is true. The three propositions (X • Y), (~X • Y), and (X • ~Y) are mutually exclusive, since no two of them can be true at the same time; but they are not exhaustive, since (~X • ~Y) is a possibility that is left out. However, all four of those propositions—taken together—are a mutually exclusive and exhaustive set, so their probabilities sum to 1:

pr(X • Y) + pr(~X • Y) + pr(X • ~Y) + pr(~X • ~Y) = 1

Similarly, a proposition and its negation are mutually exclusive and exhaustive, so their probabilities must sum to 1:

pr(X) + pr(~X) = 1

Truth tables can be used to determine whether a set of propositions is mutually exclusive and/or exhaustive. Propositions are mutually exclusive if and only if there is no row of the truth table in which more than one of the propositions is true. The propositions are exhaustive if and only if there is no row in which all of them are false.

The truth assignment method can also be used to determine whether a set of propositions is exhaustive. The strategy here is essentially the converse of the truth assignment test for logical consistency. Begin by assigning “0” to the main connective of each proposition (rather than assigning “1” to the main connectives, as done in the test for consistency). Then calculate the truth values of any other connectives and sentence letters that can be determined based on that assumption. If some letters cannot be calculated, try all possible combinations of values for those letters. The propositions are exhaustive if and only if there is no way to assign truth values to the letters while keeping all of the propositions false.

Here are a few more examples to illustrate the rules of probability:

pr(A ∨ ~A) = 1 according to rule 1, since (A ∨ ~A) is a tautology.

pr(A • ~A) = 0 according to rule 2, since (A • ~A) is a contradiction.

If pr(A) = .6 and pr(B) = .7, then pr(A ∨ B) ≥ .7 according to rule 3, since pr(A ∨ B) must be greater than or equal to the probabilities of A and B.

If pr(A) = .6 and pr(B) = .7, then pr(A • B) ≤ .6 according to rule 4, since pr(A • B) must be less than or equal to the probabilities of A and B.

If pr(A) = .2, then pr(~A) = .8 according to rule 5. Since A and ~A are mutually exclusive and exhaustive, their probabilities must sum to 1.

The mathematics of probability is known as the probability calculus, and it has many applications. Probabilities can be used to represent objective chances, statistical frequencies, and certain kinds of evidential symmetries. Most importantly for our purposes, probabilities can also be used to represent degrees of belief (or degrees of confidence).

Probabilities that represent degrees of belief are called subjective probabilities or credences. Higher credences represent greater confidence that a proposition is true; lower credences represent lower confidence that it is true (and higher confidence that it is false). For example, your credence in proposition X is 1 when you are absolutely certain that X is true, 0 when you are certain that X is false, and ½ when you have no idea whether X is true or false.

Credences can be characterized—and, according to some philosophers, they can even be defined and measured—in terms of “betting ratios.” Your credence in proposition X is the fraction of a dollar you’d regard as the fair price for a lottery ticket that wins $1 if X is true, regardless of whether you are buying or selling the ticket. For example, if your credence in X is ¾ (i.e., you are 75% confident that X is true), then you shouldn’t pay more than 75 cents for a ticket that wins $1 when X is true. Conversely, if you are selling the ticket, you shouldn’t accept any offer less than 75 cents.

Similarly, conditional credences can be characterized in terms of conditional bets. Your conditional credence in X given Y is the fraction of a dollar you’d regard as the fair price for a lottery ticket that will be cancelled (and your money refunded) unless Y is true, in which case the ticket wins $1 if X is also true.