30.1 VENN DIAGRAMS
A B
S
1
(^23)
4
5
6
Figure 30.2 The Venn diagram for the outcomes of the die-throwing trials
described in the worked example.A
AA
AB
BB
S
SS
S
(a) (b)(c) (d)A ̄
Figure 30.3 Venn diagrams: the shaded regions show (a)A∩B,theinter-
section of two eventsAandB,(b)A∪B, the union of eventsAandB,(c)
the complementA ̄of an eventA,(d)A−B, those outcomes inAthat do not
belong toB.graphically explicit the fact thatAandBare disjoint. It is not necessary, however,
to draw the diagram in this way, since we may simply assign zero outcomes to
the shaded region in figure 30.3(a). An event that contains no outcomes is called
theempty eventand denoted by∅. The event comprising all the elements that
belong to eitherAorB, or to both, is called theunionofAandBand is denoted
byA∪B(see figure 30.3(b)). In the previous example,A∪B={ 2 , 3 , 4 , 6 }.
It is sometimes convenient to talk about those outcomes that donotbelong to
a particular event. The set of outcomes that do not belong toAis called the
complementofAand is denoted byA ̄(see figure 30.3(c)); this can also be written
asA ̄=S−A. It is clear thatA∪A ̄=SandA∩A ̄=∅.
The above notation can be extended in an obvious way, so thatA−Bdenotesthe outcomes inAthat do not belong toB. It is clear from figure 30.3(d) that
A−BcanalsobewrittenasA∩B ̄. Finally, whenallthe outcomes in eventB
(say) also belong to eventA, butAmay contain, in addition, outcomes that do