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24 CHAPTER 2 PROBABILITY

A ∩ B
A B A B A B

S S S

(a) (b) (c)
Sample space S with events A and B
(A ∪ B) ∩ C (A ∩ C)'
A B A B

S S
C C

(d) (e)
Figure 2-8 Venn diagrams.

Diagrams are often used to portray relationships between sets, and these diagrams are
also used to describe relationships between events. We can use Venn diagrams to 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 A and B are
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.

Definition
Two events, denoted as E and E , such that
1
2
¨ E
E 1 2
are said to be mutually exclusive.

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

1E¿2¿ E

The distributive law for set operations implies that

1A ´ B2 ¨ C 1A ¨ C2 ´ 1B ¨ C2, and 1A ¨ B2 ´ C 1A ´ C2 ¨ 1B ´ C2