PROBABILITY
A
B
C
S
1
2
3
(^45)
6
7
8
Figure 30.4 The general Venn diagram for three events is divided into eight
regions.not belong toB,thenBis called asubsetofA, a situation that is denoted by
B⊂A; alternatively, one may writeA⊃B, which states thatAcontainsB.Inthis
case, the closed curve representing the eventBis often drawn lying completely
within the closed curve representing the eventA.
The operations∪and∩are extended straightforwardly to more than twoevents. If there existneventsA 1 ,A 2 ,...,An, in some sample spaceS, then the
event consisting of all those outcomes that belong toone or moreof theAiis the
unionofA 1 ,A 2 ,...,Anand is denoted by
A 1 ∪A 2 ∪···∪An. (30.1)Similarly, the event consisting of all the outcomes that belong toevery oneof the
Aiis called theintersectionofA 1 ,A 2 ,...,Anand is denoted by
A 1 ∩A 2 ∩···∩An. (30.2)If, foranypair of valuesi, jwithi=j,
Ai∩Aj=∅ (30.3)then the eventsAiandAjare said to bemutually exclusiveordisjoint.
Consider three eventsA,BandCwith a Venn diagram such as is shown infigure 30.4. It will be clear that, in general, the diagram will be divided into eight
regions and they will be of four different types. Three regions correspond to a
single event; three regions are each the intersection of exactly two events; one
region is the three-fold intersection of all three events; and finally one region
corresponds to none of the events. Let us now consider the numbers of different
regions in a generaln-event Venn diagram.
For one-event Venn diagrams there are two regions, for the two-event casethere are four regions and, as we have just seen, for the three-event case there are
eight. In the generaln-event case there are 2nregions, as is clear from the fact
that any particular regionRlies either inside or outside the closed curve of any
particular event. With two choices (inside or outside) for each ofnclosed curves,
there are 2ndifferent possible combinations with which to characteriseR.Oncen