Probability and Randomness

Random outcomes, in ordinary parlance, are those that occur haphazardly, unpredictably, or by chance. Even without further clarification, these glosses suggest an interesting connection between randomness and probability, in some of its guises. But we need to be more precise to articulate the relationship between the two subjects of our title. There is a large literature distinguishing kinds and analyses of probability. The corresponding philosophical literature on randomness is smaller and less familiar, although its origins lie in some of the most influential early twentieth century work on the interpretation of probability, particularly in von Mises' (1957) version of frequentism. We will begin by discussing von Mises' use of randomness in his account of physical probability 2 (§2), and the mathematical development of algorithmic randomness springing from von Mises' work (§3). We will then turn a critical eye on the relationship between randomness so conceived and physical probability, or chance (§4), and finally turn to the relationship between randomness and Humean theories of chance (§5). Though the notion of randomness has some interesting connections with evidential probability, particularly Solomonoff's version of objective Bayesianism (§3.1), for reasons of space we will focus on randomness and physical probability in this chapter. 1. Different concepts of randomness: product and process Before turning to von Mises, it will be useful to pay attention to an initial incongruence between probability and randomness. The things that can be said to have probabilities are outcomes— propositions or possible events (sometimes, sentences). But we often wish to ascribe randomness, not to these possible outcomes themselves, but to the process that produces or generates these outcomes. (So while the outcome 'Heads' bears the probability, the coin tossing process may be described as random.) We might usefully regiment at this point, and propose that the random processes are those such that at least some of the outcomes they may produce have non-trivial probabilities. (Here, 'probability' presumably means some sort of physical probability; a process that generates outcomes in which we have non-trivial credences is one that we might believe to be random in this sense, but which may not be.) In making this sensible regimentation, we follow proposals like this: I group random with stochastic or chancy, taking a random process to be one which does not operate wholly capriciously or haphazardly but in accord with stochastic or probabilistic laws. (Earman, 1986: 137) 3 But this isn't the only sense …