Given the definition of events, let us consider how we may build possibly complex events that have a practical relevance. Indeed, we often deal with the following concepts:
- The probability that an event does not occur
- The probability that at least one of two events occurs
- The probability that two events occur jointly
Since events are sets, it is natural to translate the concepts above in terms of set theory, relying on the usual difference, union, and intersection of sets.
The difference between sets A and B, denoted by A\B, is a set consisting of the elements of A that do not belong to B. Given an event E ⊆ Ω, its complement E⊂ ≡ Ω\E occurs if and only if E does not. For instance, in dice throwing we have EVEN⊂ = ODD. Graphically, the complement of an event can be depicted as in Fig. 5.2. The probability of the complement E⊂ is just the probability that event E does not occur, but these two probabilities should be related in a plausible way.
The union of two sets A and B, denoted by A ∪ B, is a set, consisting of the elements that belong to at least one of them (either A, or B, or both). Set union is depicted in Fig. 5.3. We immediately see that the probability of the union of two events is the probability that at least one of them (possibly both) occurs. Please note that we are not requiring that exactly one of them occurs. That would be an exclusive OR operation, which is perfectly legitimate per se, but set union is based on an inclusive OR operation.
Finally, the intersection of two sets A and B, denoted by A ∩ B, is a set, consisting of the elements that belong to both A and B as illustrated in Fig. 5.4. We immediately see that the probability of the intersection of two events is the probability that both of them occur jointly.2
Fig. 5.3 The union of two sets.
Fig. 5.4 The intersection of two sets.
Fig. 5.5 The disjoint of two sets.
The empty set, denoted by ø, is a set with no element. Two sets are called disjoint if their intersection is the empty set, i.e., A ∩ B = ø (see Fig. 5.5).
Given a sample space Ω, by the repeated application of these elementary set operations, we can build a huge collection of subsets of Ω. Let F the family of all sets we can build this way, working on events within a given sample space. We would like to assign probabilities to events, using sensible rules of the game, in such a way that the probabilities of complicated events are consistently related to the probabilities of the events that we used to build them. This is where the axioms of probability theory come into play. They are described in the next section in a simple and intuitive manner. We should mention that this intuition is all we need for the rest of the book, but a proper construction of probability theory is not that trivial when we deal with infinite sample spaces and possibly infinite collections of events. Generally speaking, going to infinite is always a tricky endeavor in the realm of mathematics. Still, the intuition we build, based on finite sample spaces, is perfectly adequate to our purposes.
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