Conditional Probability

A fair die is about to be tossed. The probability that it lands with ‘5’ showing up is 1/6; this is an unconditional probability. But the probability that it lands with ‘5’ showing up, given that it lands with an odd number showing up, is 1/3; this is a conditional probability. In general, conditional probability is probability given some body of evidence or information, probability relativised to a specified set of outcomes, where typically this set does not exhaust all possible outcomes. Yet understood that way, it might seem that all probability is conditional probability — after all, whenever we model a situation probabilistically, we must initially delimit the set of outcomes that we are prepared to countenance. When our model says that the die may land with an outcome from the set {1, 2, 3, 4, 5, 6}, it has already ruled out its landing on an edge, or on a corner, or flying away, or disintegrating, or . . . , so there is a good sense in which it is taking the non-occurrence of such anomalous outcomes as “given”. Conditional probabilities, then, are supposed to earn their keep when the evidence or information that is “given” is more specific than what is captured by our initial set of outcomes. In this article we will explore various approaches to conditional probability, canvassing their associated mathematical and philosophical problems and numerous applications. Having done so, we will be in a better position to assess whether conditional probability can rightfully be regarded as the fundamental notion in probability theory after all. Historically, a number of writers in the pantheon of probability took it to be so. Johnson [1921], Keynes [1921], Carnap [1952], Popper [1959b], Jeffreys [1961], Renyi [1970], and de Finetti [1974/1990] all regarded conditional probabilities as primitive. Indeed, de Finetti [1990, 134] went so far as to say that “every prevision, and, in particular, every evaluation of probability, is conditional; not only on the mentality or psychology of the individual involved, at the time in question, but also, and especially, on the state of information in which he finds himself at that moment”. On the other hand, orthodox probability theory, as axiomatized by Kolmogorov [1933], takes unconditional probabilities as primitive and later analyses conditional probabilities in terms of them. Whatever we make of the primacy, or otherwise, of conditional probability, there is no denying its importance, both in probability theory and in the myriad applications thereof — so much so that the author of an article such as this faces hard choices of prioritisation. My choices are targeted more towards a philosophical audience, although I hope that they will be of wider interest as well.