# a description of the important mathematical concepts

Equation (1) is fundamental for everything that follows. Indeed, in the modern axiomatic theory of probability, which eschews a definition of probability in terms of “equally likely outcomes” as being hopelessly circular, an extended form of equation (1) plays a basic role (*see* the section Infinite sample spaces and axiomatic probability).

An elementary, useful consequence of equation (1) is the following. With each event *A* is associated the complementary event *A*^{c} consisting of those experimental outcomes that do not belong to *A*. Since *A* ∩ *A*^{c} = Ø, *A* ∪ *A*^{c} = *S*, and *P*(*S*) = 1 (where *S* denotes the sample space), it follows from equation (1) that *P*(*A*^{c}) = 1 − *P*(*A*). For example, the probability of “at least one head” in *n* tosses of a coin is one minus the probability of “no head,” or 1 − 1/2^{n}.

This last example illustrates the fundamental principle that, if the event whose probability is sought can be represented as the union of several other events that have no outcomes in common (“at most one head” is the union of “no heads” and “exactly one head”), then the probability of the union is the sum of the probabilities of the individual events making up the union. To describe this situation symbolically, let *S* denote the sample space. For two events *A* and *B*, the intersection of *A* and *B* is the set of all experimental outcomes belonging to both *A* and *B* and is denoted *A* ∩ *B*; the union of *A* and *B* is the set of all experimental outcomes belonging to *A*or *B* (or both) and is denoted *A* ∪ *B*. The impossible event—i.e., the event containing no outcomes—is denoted by Ø. The probability of an event *A* is written *P*(*A*). The principle of addition of probabilities is that, if *A*_{1}, *A*_{2},…, *A*_{n}are events with *A*_{i} ∩ *A*_{j} = Ø for all pairs *i* ≠ *j*, then

Equation (1) is consistent with the relative frequency interpretation of probabilities; for, if *A*_{i} ∩ *A*_{j} = Ø for all *i* ≠ *j*, the relative frequency with which at least one of the *A*_{i} occurs equals the sum of the relative frequencies with which the individual *A*_{i} occur.