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 A1, A2,…, An are events with Ai ∩ Aj = Ø for all pairs i ≠ j, then
Images
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Equation (1) is consistent with the relative frequency interpretation of probabilities; for, if Ai ∩ Aj = Ø for all i ≠ j, the relative frequency with which at least one of the Ai occurs equals the sum of the relative frequencies with which the individual Ai occur.
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 Ac consisting of those experimental outcomes that do not belong to A. Since A ∩ Ac = Ø, A ∪ Ac = S, and P(S) = 1 (where S denotes the sample space), it follows from equation (1) that P(Ac) = 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 − 1/2n.
Images
Equation (1) is consistent with the relative frequency interpretation of probabilities; for, if Ai ∩ Aj = Ø for all i ≠ j, the relative frequency with which at least one of the Ai occurs equals the sum of the relative frequencies with which the individual Ai occur.
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 Ac consisting of those experimental outcomes that do not belong to A. Since A ∩ Ac = Ø, A ∪ Ac = S, and P(S) = 1 (where S denotes the sample space), it follows from equation (1) that P(Ac) = 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 − 1/2n.