ABSTRACT
Probability is a measure for the frequency of occurrence of an event. Intuitively, in an experiment this can be explained as the ratio of the number of favorable events to the number of possible outcomes. However, a somewhat more stringent definiton is helpful for a rigorous mathematical foundation (Kolmogorov, see e.g. Papoulis 1984). Axiomatically, this is described by events related to setsA,B, . . . contained in the set, which is the set of all possible events, and a non-negativemeasureProb(i.e. Probability) defined on these sets following three axioms:
I : 0 ≤ Prob[A] ≤ 1 II : Prob[] = 1 (2.1) III : Prob[A ∪ B] = Prob[A]+ Prob[B]
Axiom III holds if A and B are mutually exclusive, i.e. A∩B=∅. N.B: The probability associated with the union of two nonmutually exclusive events
(cf. the example ofA and C shown in Fig. 2.1) is not equal to the sum of the individual probabilities, Prob[A∪ C] =Prob[A]+Prob[C]. In this example there is an apparent overlap of the two events defined by A∩ C. By removing this overlap, we again obtain mutually exclusive events. From this argument we obtain:
Prob[A ∪ C] = Prob[A]+ Prob[C]− Prob[A ∩ C] (2.2) Given an event A within the set of all possible events we can define the complementary Event A¯=\A (see Fig. 2.2). Obviously, A and A¯ are mutually exclusive, hence:
=
because of Prob[A∩ A¯]=Prob[∅]=0. It can be noted that an impossible event has zero probability but the reverse is not necessarily true.
