3.2Sample Space and Events 57
In Example 1 ifE ={g}, thenEis the event that the child is a girl. Similarly, if
F={b}, thenFis the event that the child is a boy.
In Example 2 if
E={all outcomes inSstarting with a 3}thenEis the event that the number 3 horse wins the race.
For any two eventsEandFof a sample spaceS, we define the new eventE∪F, called
theunionof the eventsEandF, to consist of all outcomes that are either inEor inFor in
bothEandF. That is, the eventE∪Fwill occur ifeither EorFoccurs. For instance, in
Example 1 ifE={g}andF={b}, thenE∪F={g,b}. That is,E∪Fwould be the
whole sample spaceS. In Example 2 ifE={all outcomes starting with 6}is the event that
the number 6 horse wins andF={all outcomes having 6 in the second position}is the
event that the number 6 horse comes in second, thenE∪Fis the event that the number
6 horse comes in either first or second.
Similarly, for any two eventsEandF, we may also define the new eventEF, called the
intersectionofEandF, to consist of all outcomes that are in bothEandF. That is, the
eventEFwill occur only if bothEandFoccur. For instance, in Example 3 ifE=(0, 5)
is the event that the required dosage is less than 5 andF=(2, 10) is the event that it is
between 2 and 10, thenEF =(2, 5) is the event that the required dosage is between 2
and 5. In Example 2 ifE ={all outcomes ending in 5}is the event that horse number
5 comes in last andF={all outcomes starting with 5}is the event that horse number 5
comes in first, then the eventEFdoes not contain any outcomes and hence cannot occur.
To give such an event a name, we shall refer to it as the null event and denote it by∅.
Thus∅refers to the event consisting of no outcomes. IfEF=∅, implying thatEandF
cannot both occur, thenEandFare said to bemutually exclusive.
For any eventE, we define the eventEc, referred to as thecomplementofE, to consist
of all outcomes in the sample spaceSthat are not inE. That is,Ecwill occur if and only
ifEdoes not occur. In Example 1 ifE={b}is the event that the child is a boy, then
Ec={g}is the event that it is a girl. Also note that since the experiment must result in
some outcome, it follows thatSc=∅.
For any two eventsEandF, if all of the outcomes inEare also inF, then we say that
Eis contained inFand writeE⊂F(or equivalently,F⊃E). Thus ifE⊂F, then the
occurrence ofEnecessarily implies the occurrence ofF.IfE⊂FandF⊂E, then we say
thatEandFare equal (or identical) and we writeE=F.
We can also define unions and intersections of more than two events. In particu-
lar, the union of the eventsE 1 ,E 2 ,...,En, denoted either byE 1 ∪E 2 ∪ ··· ∪Enor by
∪n 1 Ei, is defined to be the event consisting of all outcomes that are inEifor at least one
i=1, 2,...,n. Similarly, the intersection of the eventsEi,i=1, 2,...,n, denoted by
E 1 E 2 ···En, is defined to be the event consisting of those outcomes that are in all of the
eventsEi,i=1, 2,...,n. In other words, the union of theEioccurs whenat leastone of
the eventsEioccurs; the intersection occurs whenallof the eventsEioccur.