Foundations of the theory of probability

4 I. ElementaryTheoryofProbability

uponrealizationofconditions 8 belongstothesetA (definedin

anyway),thenwesaythattheeventAhastakenplace.

Example:Letthecomplex

3

ofconditionsbethetossingofa

cointwotimes.ThesetofeventsmentionedinParagraph^con-

sistsofthefactthatateachtosseither
a

headortailmaycomeup.

From
thisitfollowsthat

only

four

differentvariants (elementary

events) arepossible,namely:HH,HT,TH,TT.Ifthe"event

A"

connotestheoccurrence ofarepetition, thenitwillconsistofa

happeningofeitherofthefirstorfourthofthefourelementary

events.Inthismanner,everyeventmayberegardedasasetof

elementaryevents.

4)

Undercertainconditions,which

we

shallnotdiscusshere,

wemay

assumethattoaneventAwhichmayormaynotoccur

underconditions8, is assignedareal numberP(A) whichhas

thefollowingcharacteristics

:

(a) Onecanbepracticallycertainthatifthecomplexofcon-

ditions

6

isrepeatedalargenumberoftimes,n,thenifmbethe

numberofoccurrencesofeventA,

theratiom/n

will

differvery

slightlyfromP

(

A

)

.

(b) IfP(A) isverysmall,onecanbepracticallycertainthat

whenconditions@arerealizedonlyonce,theeventAwouldnot

occuratall.

TheEmpirical

DeductionoftheAxioms.In

general,onemay

assumethatthesystem

g

oftheobservedeventsA,B,C, ...

to

whichareassigneddefiniteprobabilities,formafield

containing

as an element the set E (Axioms I, II, and the first part of

III, postulating theexistence of probabilities). Itis clearthat

O^m/n^l

so

thatthesecondpartofAxiomIIIisquitenatural.

For

theeventE,misalways

equal

ton,so

thatitisnaturalto

postulate ?(E)

=

(Axiom IV).

If, finally,

A andB

are non-

intersecting (incompatible),thenm

—

m

1

+m

2

wherem,m

lt

m

2

arerespectivelythenumberofexperimentsinwhichtheevents

A +B,A,andBoccur.Fromthisitfollowsthat

m m

1

m

2

n n n

Ittherefore

seems

appropriate

to

postulate that P(A +B)

—

P(A) +P(J5) (Axiom V).