Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

24 Probability and Distributions

Moreover, from a relative frequency point of view, it would seem logically incon-

sistent if we did not require that the ratio of the probabilities of the eventsA∩B

andA, relative to the spaceA, be the same as the ratio of the probabilities of these

events relative to the spaceC; that is, we should have

P(A∩B|A)

P(A|A)

=

P(A∩B)

P(A)

.

These three desirable conditions imply that the relation conditional probability is
reasonably defined as

Definition 1.4.1(Conditional Probability).LetBandAbe events withP(A)> 0.
Then we defined theconditional probabilityofBgivenAas

P(B|A)=

P(A∩B)

P(A)

. (1.4.1)

Moreover, we have

1.P(B|A)≥0.

2.P(A|A)=1.

3.P(∪∞n=1Bn|A)=

∑∞

n=1P(Bn|A),provided thatB^1 ,B^2 ,...are mutually ex-

clusive events.

Properties (1) and (2) are evident. For Property (3), suppose the sequence of
eventsB 1 ,B 2 ,...is mutually exclusive. It follows that also (Bn∩A)∩(Bm∩A)=φ,
n =m. Using this and the first of the distributive laws (1.2.5) for countable unions,
we have

P(∪∞n=1Bn|A)=

P[∪∞n=1(Bn∩A)]

P(A)

=

∑∞

n=1

P[Bn∩A]

P(A)

=

∑∞

n=1

P[Bn|A].

Properties (1)–(3) are precisely the conditions that a probability set function must
satisfy. Accordingly,P(B|A) is a probability set function, defined for subsets ofA.
It may be called the conditional probability set function, relative to the hypothesis
A, or the conditional probability set function, givenA. It should be noted that
this conditional probability set function, givenA, is defined at this time only when
P(A)>0.

Example 1.4.1.A hand of five cards is to be dealt at random without replacement
from an ordinary deck of 52 playing cards. The conditional probability of an all-
spade hand (B), relative to the hypothesis that there are at least four spades in the