Basics of Probability Theory
PROBABILITY AND RANDOM PHENOMENA
Probability theory is concerned with the study of random phenomena, that is, those lacking deterministic regularity, while possessing at least some degree of statistical regularity. This means that while observations of random phenomena cannot be predicted with certainty, the observed frequency of the outcomes can in some sense be foretold. As is often the case in phenomenology, the qualifiers “with certainty” and “in some sense” are subject to interpretation, and the perceived deterministic or random nature of a particular phenomenon can depend on the meaning assigned to these qualifiers. One could say, in fact, that since, with a pair of well-known exceptions, nothing in life is certain, all phenomena are to some degree random.
In order to illustrate these concepts, let us examine that most prototypical of random phenomena, the tossing of a coin. In most people's mind a toss of a coin can have one of two equally likely outcomes: heads or tails. The expectation of equal likelihood derives from intuition and, while usually taken as obvious, relies on a number of implicit assumptions. Leaving aside the unlikely but not inconceivable possibility of a third outcome (e.g., the coin could roll into a crack and become lost or wedged in a vertical position), these include (1) the coin being fair, or unbiased; (2) the tosses not being mechanically identical; and (3) the trials being independent. In practice, one ...