Conditional Probability

Conditional Probability is a concept used frequently in probability theory and thus also in probability-based theories of conditionals. Conditional probability is the probability of some claim being true under the assumption that some (other) claim is true.

Compare these two claims: (i) I will finish my homework today (ii) I will finish my homework today, given that I am not sick. Suppose yesterday I said (i) and I did not finish my homework. Then, it must be that I spoke falsly. But suppose yesterday I said (ii) instead and I did not finish my homework, it's possible I did not speak falsely. This is a difference between what can be called an assertive claim and a conditional claim respectively.

Corresponding these two kinds of claims, we can say about the probability that I finish my homework today and the probability that I finish my homework today, given that I am not sick. The former is absolute or categorical probability and the latter conditional probability. So the conditional probability has the form of the probability of C, given A, and is written as Pr(C|A), where '|' stands for 'given'. Usually the conditional probability is defined in terms of the ordinary (non-conditional) probability in what is called the ratio formula:

Pr(C|A) = Pr(C&A) / Pr(A)

According to the ratio formula, the probability of C given A is the probability of A&C divided by the probability of A. This equation only holds when the probability of A is non-zero. The equation also allows us to calculate the probability of A&C with the probability of A and the probability of C given A. In practice, one usually uses knowledge of Pr(A) and Pr(C|A) to calculate P(C&A).

Reducibility of Conditional Probability to Unconditional Probability

Although Pr(C|A) can be defined formally as the ratio Pr(C&A) / Pr(A), it is controversial whether it is right to regard the equation as a definition or an analysis of the conditional probability. Kolmogorov, who first axiomatized our concept of probability, offered the equation as the definition of the conditional probability. (See Kolmogorov, A. N. (1933), Grundbegriffe der Wahrscheinlichkeitrechnung, tanslated as Foundation of Probability (1950), Chelsea Publishing Company, New York.) On the other hand, some philosophers contend that the equation is rather a kind of an "analysis" of conditional probability. For example, Alan Hájek calls the ratio formular "the ratio analysis of conditional probability", and argues that the analysis fails to be an adequate analysis of conditional probability. (See Hájek (2003), "What Conditional Probability Could Not Be", Synthese 137: pp. 273-323.) According to him, conditional probability should be taken as the primitive notion in probability theory, and unconditional probability should be analyzed in terms of it.

Here are two arguments for the irreducibility of conditional probability to the ratio.

Pr(C|A) is defined when P(A) or P(A&C) is Undefined

In order to have a subjective probability of C given A, it is not necessary to have a probability for A or A&C. For example, even if I don't have any belief about how likely it is that the lecture will be cancelled, I may still a high degree of belief that there will be no refreshments served, given that the lecture is cancelled.

Pr(C|A) is defined when P(A)=0

Consider a dartboard with a horizontal line segment drawn on it. Because the line segment has no thickness the probability of a dart hitting the line segment is zero, but there is a conditional probability that is still defined. The probability of the dart landing on the left third of the line segment given that the dart lands on the line segment is 1/3.