Relation of Probability to Randomness

Probability is not necessarily related to randomness but manifests itself in all types of phenomena, deterministic or stochastic.

Probability is nowadays defined axiomatically but in the past it was defined in terms of relative frequency or equally likely alternatives.  The axiomatic definition of probability is plagued by the dubious assumption of infinite sigma additivity, which constitutes its third axiom. Although this axiom can be relaxed to finite additivity, this often leads to violations of the first axiom and negative probabilities.  Negative probabilities are encountered in quantum mechanics but their notion is counter-intuitive. Thus, there is actually not one axiomatic definition but several depending on the form of the third axiom. It is very hard to conceive how negative probabilities in the microcosm give rise to strictly positive probabilities in the macrocosm, unless the two cosmos are related is some weird way. This casts a dark cloud over the speculation that reality is inherently probabilistic and raises questions about probability being a tool that “compensates for imprecise knowledge”.

The relative frequency definition of probability used by engineers mainly assumes that a limit exists as the number of repetitions of an experiment becomes infinite. For example, as the number of tosses of a fair coin becomes very large, the probability of heads is the limit of the relative frequency of the occurrence of heads.  However, repetitions will always be finite in the real world and thus an assumption is made that the limit exists as their number becomes infinite. This assumption is not bad in many practical cases, as technological advancements prove posteriori.

The classical definition of probability, given as the ratio of favorable to the total number of outcomes, assumes that the outcomes are equally likely and it is the one that reinforced the association between probability and randomness in people’s minds. “Equally likely” already means equally probable. Such definition is circular and would make sense if the underline process of selecting the outcomes is random. However, the association is false because underneath this definition of equally likely outcomes hides the relative frequency interpretation of probability: we call “equally likely” outcomes that at the limit have equal probabilities and not those that result from some hypothetical random selection process.

Just because there is a probability of a call in a time interval, or that the probability of heads in a coin toss is p, that does not directly imply that the world is random. Probability does not imply randomness although the opposite is true, i.e. randomness is a sufficient condition for probability. Given this sufficiency and the paradox of material implication, any proof that this world is random because of the existence of probabilistic processes cannot be established because, due to the paradox, whether the world is random or not, it is true that such processes are present.

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