Foundations of the theory of probability

§

2.TheRelationtoExperimental

Data

3

Oursystem

ofaxiomsisnot,however,complete,forin

various

problems in the

theory of probabilitydifferent fieldsof proba-

bilityhavetobeexamined.

TheConstruction

of

Fields

of

Probability.Thesimplestfields

ofprobabilityareconstructedas

follows.Wetakeanarbitrary

finitesetE

=

{|

t

,£

2

,

..

.,£*}

andanarbitraryset

{p

ltp 2

,

..

.,

pk

)

ofnon-negativenumbers withthesum
Pi

+

p 2

+•• •+

Pk

—

1.

g

istakenasthesetofallsubsetsinE,andweput

P{ft

i

,^,...,^}

=

^

i

+

fc

+

• 
  

v

+^.

Insuchcases,
p

up 2 ,

...
,
p

k

arecalledthe

probabilities

of

the

elementaryevents
$

1}£ 2 ,

...

,$k

orsimplyelementary

probabili-

ties.Inthiswayarederivedallpossible
finite

fieldsofprobability

inwhich
gf

consists ofthesetofallsubsetsofE. (Thefieldof

probability
is

called

finite

if the set E is finite.) For further

examplesseeChap.II,
§

3.

§

2. TheRelationtoExperimental

Data

4

We apply the theory of probabilityto the actual world

of

experimentsinthefollowingmanner:

1)

Thereisassumedacomplexofconditions,

©,

whichallows

ofanynumberofrepetitions.

2) Westudyadefinitesetofevents

which

couldtake

placeas

aresultofthe establishmentofthe conditionsS. In

individual

caseswheretheconditionsarerealized,theeventsoccur,

gener-

ally,indifferentways. LetEbethesetofallpossiblevariants

d,&,

...oftheoutcomeofthegivenevents.Someofthesevari-

antsmightingeneralnotoccur.WeincludeinsetEallthevari-

antswhichweregardaprioriaspossible.

3)

Ifthevariantof theevents which

hasactuallyoccurred

4

Thereaderwhoisinterested

inthe

purely

mathematical

developmentof

thetheoryonly,neednotreadthissection,since

the

workfollowing

itis

based

onlyupontheaxiomsin

§

1 andmakes

nouseofthe

presentdiscussion.Here

welimitourselves

toa

simpleexplanation

ofhowtheaxiomsofthetheory

of

probability

aroseanddisregardthedeepphilosophicaldissertationsonthe

conceptofprobabilityintheexperimentalworld.Inestablishingthepremises

necessary

fortheapplicabilityofthetheoryofprobabilitytotheworldof

actualevents,theauthorhasused,inlargemeasure,theworkofR.v.Mises,

[1]pp.

21-27.