Mathematical Methods for Physics and Engineering - Matematica.NET

30.2 PROBABILITYtimes then we expect that a six will occur approximately N/6 times (assuming,of course, that the die is not biased). The regularity of outcomes allows us todefine the probability, Pr(A), as the expected relative frequency of event A in alarge number of trials. More quantitatively, if an experiment has a total of n Soutcomes in the sample space S, andn A of these outcomes correspond to theevent A, then the probability that event A will occur isPr(A) = n A. (30.5)n S30.2.1 Axioms and theoremsFrom (30.5) we may deduce the following properties of the probability Pr(A).(i) For any event A in a sample space S,0 ≤ Pr(A) ≤ 1. (30.6)If Pr(A) =1thenA is a certainty; if Pr(A) =0thenA is an impossibility.(ii) For the entire sample space S we havePr(S) = n S=1, (30.7)n Swhich simply states that we are certain to obtain one of the possibleoutcomes.(iii) If A and B are two events in S then, from the Venn diagrams in figure 30.3,we see thatn A∪B = n A + n B − n A∩B , (30.8)the final subtraction arising because the outcomes in the intersection ofA and B are counted twice when the outcomes of A are added to thoseof B. Dividing both sides of (30.8) by n S , we obtain the addition rule forprobabilitiesPr(A ∪ B) =Pr(A)+Pr(B) − Pr(A ∩ B). (30.9)However, if A and B are mutually exclusive events (A ∩ B = ∅) thenPr(A ∩ B) = 0 and we obtain the special casePr(A ∪ B) =Pr(A)+Pr(B). (30.10)(iv) If Ā is the complement of A then Ā and A are mutually exclusive events.Thus, from (30.7) and (30.10) we have1=Pr(S) =Pr(A ∪ Ā) =Pr(A)+Pr(Ā),from which we obtain the complement lawPr(Ā) =1− Pr(A). (30.11)1125