Chapter 1
Classical Probabilit y
Probability does not exist.
-Bruno d e Finett i
How dare we speak of the laws of chance? Is not chance the antithesis of all law?
-Bertrand Russel l
The origina l developmen t o f probabilit y theor y too k plac e durin g th e
seventeenth throug h nineteent h centuries . I n thos e time s th e subjec t wa s
mainly concerne d wit h game s o f chance . Sinc e the n a n increasin g numbe r o f
scientific applications , man y i n mathematic s itself , hav e spurre d th e devel -
opment o f probabilit y i n othe r directions . Nonetheless , classica l probabilit y
remains a mos t natura l plac e t o star t th e subject . I n thi s wa y w e ca n wor k
non-axiomatically an d loosel y i n orde r t o gras p a numbe r o f usefu l idea s
without havin g t o develo p abstrac t machinery . Tha t wil l b e covere d i n late r
chapters.
1. Discret e Probabilit y
Consider a gam e tha t ca n lea d t o a fixed denumerabl e (i.e. , a t mos t count -
able) se t { ^ } ^
1
o f possible outcomes . Suppos e i n additio n tha t th e outcom e
ujj occur s wit h probabilit y pj (j = 1, 2 , . . . ), wher e w e agre e onc e an d fo r al l
tha t probabilitie s ar e rea l number s i n [0,1], an d tha t th e u^' s reall y ar e th e
only possibl e outcomes ; i.e. , tha t Yli^iPj
=
1-
1
http://dx.doi.org/10.1090/gsm/080/01
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