Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.2 PROBABILITY

times then we expect that a six will occur approximatelyN/6 times (assuming,

of course, that the die is not biased). The regularity of outcomes allows us to

define theprobability,Pr(A), as the expected relative frequency of eventAin a

large number of trials. More quantitatively, if an experiment has a total ofnS

outcomes in the sample spaceS,andnAof these outcomes correspond to the

eventA, then the probability that eventAwill occur is

Pr(A)=

nA

nS

. (30.5)

30.2.1 Axioms and theorems

From (30.5) we may deduce the following properties of the probability Pr(A).

(i) For any eventAin a sample spaceS,

0 ≤Pr(A)≤ 1. (30.6)

If Pr(A)=1thenAis a certainty; if Pr(A)=0thenAis an impossibility.

(ii) For the entire sample spaceSwe have

Pr(S)=

nS

nS

=1, (30.7)

which simply states that we are certain to obtain one of the possible

outcomes.

(iii) IfAandBare two events inSthen, from the Venn diagrams in figure 30.3,

we see that

nA∪B=nA+nB−nA∩B, (30.8)

the final subtraction arising because the outcomes in the intersection of

AandBare counted twice when the outcomes ofAare added to those

ofB. Dividing both sides of (30.8) bynS, we obtain theaddition rulefor

probabilities

Pr(A∪B)=Pr(A)+Pr(B)−Pr(A∩B). (30.9)

However, ifAandBaremutually exclusiveevents (A∩B=∅)then

Pr(A∩B) = 0 and we obtain the special case

Pr(A∪B)=Pr(A)+Pr(B). (30.10)

(iv) IfA ̄is the complement ofAthenA ̄andAare mutually exclusive events.

Thus, from (30.7) and (30.10) we have

1=Pr(S)=Pr(A∪A ̄)=Pr(A)+Pr(A ̄),

from which we obtain thecomplement law

Pr(A ̄)=1−Pr(A). (30.11)