Foundations of the theory of probability

§

3. NotesonTerminology 5

Remark 1. Iftwo separate statementsare eachpractically

reliable,thenwemaysaythatsimultaneouslytheyareboth

reli-

able,

althoughthedegreeofreliabilityissomewhatloweredinthe

process.If,however,thenumberofsuchstatementsisverylarge,

thenfrom
the

practicalreliabilityofeach,onecannotdeduceany-

thing

about

thesimultaneouscorrectnessofallofthem.Therefore

fromthe

principle

statedin(a)

it

does

notfollow

that

in

a

very

largenumberofseries

of

ntestseach,

ineachtheratiom/nwill

differonlyslightlyfromP(A).

Remark 2. To an impossible event (an empty set) corre-

sponds,inaccordancewithouraxioms,theprobabilityP(0)

=

5

,

buttheconverse isnottrue: P(A) =0doesnotimplytheim-

possibilityofA.WhenP(A)

—

0,

fromprinciple (b)allwecan

assertisthatwhentheconditions

©

arerealizedbutonce,event

Aispracticallyimpossible.Itdoesnotatallassert,however,that

inasufficientlylongseriesofteststheeventAwillnot

occur.On

theotherhand,onecandeducefromthe

principle

(a)

merelythat

whenP(A)

=

andnisverylarge,theratio m/nwillbevery

small (itmight,forexample,beequalto1/n).

§

3. NotesonTerminology

We have defined the

objects of our future study, random

events,

assets.However, inthetheoryofprobabilitymanyset-

theoreticconceptsaredesignatedbyotherterms. Weshallgive

hereabrieflistofsuchconcepts.

Theory

of

Sets

Random

Events

1. AandBdonotintersect,
   
   
   1. Events A andBare
   
   
   
   

in-

i.e.,AB

—

0. compatible.


1. AB..

.2V~=

0. 
   
   
   
   2. EventsA,B,...,2Vare
   
   
   
   

incompatible.

3. AB...N

=

X.

3. EventXis

defined

as

the

simultaneous

occurrence of

eventsA,B,...,N.

4. A4-B
   
   
   • 
     
   
   
   
   

..

.+

N=X.

4. EventXisdefinedas

the

occurrence of atleast oneof

theevents

A,B,...,N.

8

Cf.§4,Formula (3).