Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.3. The Probability Set Function 13

A probability set function tells us how the probability is distributed over the set
of events,B. In this sense we speak of a distribution of probability. We often drop
the word “set” and refer toPas a probability function.
The following theorems give us some other properties of a probability set func-
tion. In the statement of each of these theorems,P(A) is taken, tacitly, to be a
probability set function defined on the collection of eventsBof a sample spaceC.

Theorem 1.3.1.For each eventA∈B,P(A)=1−P(Ac).

Proof: We haveC=A∪AcandA∩Ac=φ. Thus, from (2) and (3) of Definition
1.3.1, it follows that
1=P(A)+P(Ac),

which is the desired result.

Theorem 1.3.2.The probability of the null set is zero; that is,P(φ)=0.

Proof: In Theorem 1.3.1, takeA=φso thatAc=C. Accordingly, we have

P(φ)=1−P(C)=1−1=0

and the theorem is proved.

Theorem 1.3.3.IfAandBare events such thatA⊂B,thenP(A)≤P(B).

Proof: NowB=A∪(Ac∩B)andA∩(Ac∩B)=φ. Hence, from (3) of Definition
1.3.1,
P(B)=P(A)+P(Ac∩B).

From (1) of Definition 1.3.1,P(Ac∩B)≥0. Hence,P(B)≥P(A).

Theorem 1.3.4.For eachA∈B, 0 ≤P(A)≤ 1.

Proof: Sinceφ⊂A⊂C, we have by Theorem 1.3.3 that

P(φ)≤P(A)≤P(C)or0≤P(A)≤ 1 ,

the desired result.

Part (3) of the definition of probability says thatP(A∪B)=P(A)+P(B)ifA
andBare disjoint, i.e.,A∩B=φ. The next theorem gives the rule for any two
events regardless if they are disjoint or not.

Theorem 1.3.5.IfAandBare events inC,then

P(A∪B)=P(A)+P(B)−P(A∩B).

Proof: Each of the setsA∪BandBcan be represented, respectively, as a union of
nonintersecting sets as follows:

A∪B=A∪(Ac∩B)andB=(A∩B)∪(Ac∩B). (1.3.1)