126 PROBABILITY [CHAP. 7
7.3Finite Probability Spaces
The following definition applies.Definition 7.1: LetSbe a finite sample space, sayS={a 1 ,a 2 ,...,an}.Afinite probability space,orprobability
model, is obtained by assigning to each pointaiinSa real numberpi, called theprobabilityofaisatisfying the
following properties:
(i) Eachpiis nonnegative, that is,pi≥0.
(ii) The sum of thepiis 1, that is, isp 1 +p 2 +···+pn=1.
Theprobabilityof an eventAwrittenP (A), is then defined to be the sum of the probabilities of the points inA.
The singleton set{ai}is called anelementaryevent and, for notational convenience, we writeP(ai)
forP({ai}).
EXAMPLE 7.4 (Experiment) Suppose three coins are tossed, and the number of heads is recorded. (Compare
with the above Example 7.1(a).)
The sample space isS={ 0 , 1 , 2 , 3 }. The following assignments on the elements ofSdefine a probability
space:
P( 0 )=^18 ,P( 1 )=^38 ,P( 2 )=^38 ,P( 3 )=^18
That is, each probability is nonnegative, and the sum of the probabilities is 1. LetAbe the event that at least one
head appears, and letBbe the event that all heads or all tails appear; that is, letA={ 1 , 2 , 3 }andB={ 0 , 3 }.
Then, by definition,
P (A)=P( 1 )+P( 2 )+P( 3 )=^38 +^38 +^18 =^78 and P(B)=P( 0 )+P( 3 )=^18 +^18 =^14Equiprobable Spaces
Frequently the physical characteristics of an experiment suggest that the various outcomes of the sample
space be assigned equal probabilities. Such a finite probability spaceS, where each sample point has the same
probability, will be called anequiprobable space. In particular, ifScontainsnpoints, then the probability of
each point is 1/n. Furthermore, if an eventAcontainsrpoints, then its probability isr( 1 /n)=r/n. In other
words, wheren(A)denotes the number of elements in a set A,
P (A)=number of elements inA
number of elements inS=n(A)
n(S)or P (A)=number of outcomes favorable toA
total number of possible outcomes
We emphasize that the above formula forP (A)can only be used with respect to an equiprobable space, and
cannot be used in general.
The expressionat randomwill be used only with respect to an equiprobable space; the statement “choose
a point at random from a setS” shall mean that every sample point inShas the same probability of being
chosen.
EXAMPLE 7.5 Let a card be selected from an ordinary deck of 52 playing cards. Let
A={the card is a spade} and B={the card is a face card}.We computeP (A),P(B), andP(A∩B). Since we have an equiprobable space,
P (A)=number of spades
number of cards=13
52=1
4, P (B)=number of face cards
number of cards=12
52=3
13P(A∩B)=number of spade face cards
number of cards=3
52