We know the union, intersection and difference of sets and the complement of a set. Since events are certain subsets of the sample space, we can perform set operations on them.
The notations conventionally used for events differ from the corresponding notations of set theory. At the end of this chapter we give the set-theoretic analogues of all operations with notations.
If an event of the sample space Ω isE ⊂Ω, then thecomplement oropposite of event E is the event that occurs when E does not occur. Notation: E.
The complement of the certain event is the impossible event, and the complement of the impossible event is the certain event, i.e., Ω =∅ and ∅= Ω.
Set operations are commonly illustrated by Venn diagrams, therefore we will also use them for operations on events. Let the sample space be a rectangle Ω, and let the experiment be choosing a point of this rectangle. The complement of event E ⊂Ω is the difference Ω\E, since the complement event means choosing a point which belongs to Ω, but does not belong to E (see Fig. 1.1).
Further operations on events will also be illustrated on this experiment.
Example 1.5. In the example of rolling a die:
E :=the roll is an even number, E =the roll is an odd number, F :=the roll is a 6, F =the roll is a 1, 2, 3, 4 or 5.
Ω
E E
Figure 1.1: Complement of an event.
For a given sample space, any event and its complement (E, E) form a complete set of events consisting of two events, that is to say Ω = E∪E.
IfE andF are events in the same sample space, then thesum of E and F is defined as the event that occurs when event E or event F occurs. Notation: E +F (see Fig.
1.2).
In the definition of the sum the connective or is understood in the inclusive sense, i.e., the sum of two events can occur in three ways: either E occurs and F does not occur, or F occurs and E does not occur, or both occur. That is to say E+F occurs if at least one of event E and event F occurs.
Ω
E
F E+F
Figure 1.2: The sum of events.
We can formulate this more generally: ifE1, E2, . . . , Enare events in the same sample space, then let their sum be the event that occurs when any one of the listed events occurs.
Notation:
n
P
i=1
Ei or E1+. . .+En.
IfE1, E2, . . . form an infinite sequence of events in the same sample space, then their sum is defined as the event that occurs when any one of the listed events occurs. Notation:
∞
P
i=1
Ei orE1+E2+. . .
IfE and F are events in the same sample space, then theirproduct is the event that occurs when both events occur. The product is denoted by E ·F or, briefly, EF (see Fig. 1.3).
Ω
E
F EF
Figure 1.3: The product of events.
IfE1, . . . , Enare events in the same sample space, then let their product be the event that occurs when all the listed events occur. Notation:
n
Q
i=1
Ei orE1·. . .·En.
The events E and F are mutually exclusive if and only if EF =∅, since both mean that the two events cannot occur at the same time.
IfE andF are events in the same sample space, then thedifference of these events is defined as the event that occurs when E occurs, but F does not occur. Notation: E−F (see Fig. 1.4).
Example 1.6. In the example of rolling a die:
E :=the roll is an even number,
F :=the roll is a number that is smaller than 3, then
E+F = the roll is an even number or a number that is smaller than 3
= the roll is a 1, 2, 4 or 6,
Ω
E
F E−F
Figure 1.4: The difference of events.
EF =the roll is an even number and its value is smaller than 3 =the roll is a 2, E−F = the roll is an even number and its value is not smaller than 3
= the roll is a4 or a6.
In the following table we list the operations on events and the corresponding opera-tions on sets:
Operations on events, notations: Operations on sets, notations:
complement, E sum, E+F product, EF difference, E−F
complement, Ec union, E∪F intersection, E∩F difference, E\F
If E1, E2, . . . , En form a finite sequence of events in the same sample space, then the related de Morgan’s laws read as
E1+. . .+En =E1·. . .·En, E1·. . .·En=E1+. . .+En.
Example 1.7. In a study area three insect species (A, B, C) can be found. We place an insect trap, and after some time we look at the insects that have been caught. Let
E1 :=a specimen of species A can be found in the trap, E2 :=a specimen of species B can be found in the trap, E3 :=a specimen of species C can be found in the trap.
By using the events E1, E2, E3 and the operations, we give the following events:
(a) There is no specimen of species A in the trap=E1. There is no specimen of species B in the trap=E2. There is no specimen of species C in the trap=E3. (b) There is no insect in the trap=E1 E2 E3.
(c) There is an insect in the trap=E1+E2+E3.
(d) There are only specimens of a single species in the trap =E1 E2 E3+E1 E2 E3+ E1 E2 E3.
(e) There are specimens of two species in the trap=E1 E2 E3+E1 E2 E3+E1 E2 E3. (f ) There are specimens of all the three species in the trap=E1E2E3.
(g) Application of de Morgans’ law: There are specimens of at most two species in the trap. (There are no specimen or are specimens of one or two species but not of all the three species in the trap.) E1E2E3 =E1+E2+E3.