Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.4. Conditional Probability and Independence 23

1.3.22.Consider the eventsC 1 ,C 2 ,C 3.

(a)SupposeC 1 ,C 2 ,C 3 are mutually exclusive events. IfP(Ci)=pi,i=1, 2 ,3,

what is the restriction on the sump 1 +p 2 +p 3?

(b)In the notation of part (a), ifp 1 =4/10,p 2 =3/10, andp 3 =5/10, are

C 1 ,C 2 ,C 3 mutually exclusive?

For the last two exercises it is assumed that the reader is familiar withσ-fields.

1.3.23.SupposeDis a nonempty collection of subsets ofC. Consider the collection
of events
B=∩{E:D⊂EandEis aσ-field}.

Note thatφ∈Bbecause it is in eachσ-field, and, hence, in particular, it is in each
σ-fieldE⊃D. Continue in this way to show thatBis aσ-field.

1.3.24.LetC=R,whereRis the set of all real numbers. LetIbe the set of all
open intervals inR.TheBorelσ-field on the real line is given by

B 0 =∩{E:I⊂EandEis aσ-field}.

By definition,B 0 contains the open intervals. Because [a,∞)=(−∞,a)candB 0
is closed under complements, it contains all intervals of the form [a,∞), fora∈R.
Continue in this way and show thatB 0 contains all the closed and half-open intervals
of real numbers.

1.4 Conditional Probability and Independence

In some random experiments, we are interested only in those outcomes that are
elements of a subsetAof the sample spaceC. This means, for our purposes, that
the sample space is effectively the subsetA. We are now confronted with the
problem of defining a probability set function withAas the “new” sample space.
Let the probability set functionP(A) be defined on the sample spaceCand let
Abe a subset ofCsuch thatP(A)>0. We agree to consider only those outcomes
of the random experiment that are elements ofA; in essence, then, we takeA
to be a sample space. LetBbe another subset ofC. How, relative to the new
sample spaceA, do we want to define the probability of the eventB? Once defined,
this probability is called theconditional probabilityof the eventB,relativetothe
hypothesis of the eventA, or, more briefly, the conditional probability ofB,given
A. Such a conditional probability is denoted by the symbolP(B|A). The “|”in this
symbol is usually read as “given.” We now return to the question that was raised
about the definition of this symbol. SinceAis now the sample space, the only
elements ofBthat concern us are those, if any, that are also elements ofA,that
is, the elements ofA∩B. It seems desirable, then, to define the symbolP(B|A)in
such a way that

P(A|A)=1 and P(B|A)=P(A∩B|A).