Applied Statistics and Probability for Engineers

24 CHAPTER 2 PROBABILITY

Diagrams are often used to portray relationships between sets, and these diagrams are

also used to describe relationships between events. We can use Venn diagramsto represent a

sample space and events in a sample space. For example, in Fig. 2-8(a) the sample space of

the random experiment is represented as the points in the rectangle S. The events Aand Bare

the subsets of points in the indicated regions. Figure 2-8(b) illustrates two events with no com-

mon outcomes; Figs. 2-8(c) to 2-8(e) illustrate additional joint events.

Two events with no outcomes in common have an important relationship.

A B AB

(a)

Sample space S with events A and B

(b)

AB

(c)

AB

(e)

AB

(d)

A ∩ B

S

(A ∩ C)'

S S

(A ∪ B) ∩ C

S S

CC

Figure 2-8 Venn diagrams.

Two events, denoted as E 1 and E 2 , such that

are said to be mutually exclusive.

E 1 ̈E 2   

Definition

The two events in Fig. 2-8(b) are mutually exclusive, whereas the two events in Fig. 2-8(a)

are not.

Additional results involving events are summarized below. The definition of the comple-

ment of an event implies that

The distributive law for set operations implies that

1 A ́B 2 ̈C 1 A ̈C 2 ́ 1 B ̈C 2 , and 1 A ̈B 2 ́C 1 A ́C 2 ̈ 1 B ́C 2

1 E¿ 2 ¿E

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