Foundations of the theory of probability

I.

ElementaryTheoryof

Probability

Theory

of

Sets

5. The

complementary set

6. A

=

0.

7. A

=

E.

8. Thesystem 51 ofthesets

A

lt

A

2

,

. ..
,

A

n

formsa de-

compositionof

theset

Eif

A
1

+A

2

+...+A

n

=

E.

(This assumes that

the

sets

At

donotintersect,in

pairs.)

9. Bisasubsetof

A:2?

t

cA.

RandomEvents

5. The opposite event A

consisting of the non-occur-

ence

ofeventA.

6.

EventAisimpossible.

7. EventA

mustoccur.

8.

Experiment

%consistsof

determining which of

the

eventsA

u

A

2 ,

...

,

A

n

occurs.

WethereforecallA

l

A

2

,

..

.

,

A

n

the possibleresults ofex-

periment

51.

9. From theoccurrence

of

eventBfollowstheinevitable

occurrence ofA.

§

4. ImmediateCorollariesoftheAxioms;Conditional

Probabilities;Theorem

ofBayes

FromA

+

A

=

E

andthe

AxiomsIVandV itfollowsthat

P(A) +P(A)

=

(1)

P(A)

=1—

P(A).

(2)

SinceE

=

0,then,inparticular,

P(0)=0. (3)

IfA,B,..

.

,Nareincompatible,thenfrom

Axiom

V

follows

theformula (theAdditionTheorem)

P(A+£+... +N)=P(A) + P(£)+...+

P(N)

IfP(A)>0,thenthequotient

P(AB)

(4)

?a(B)

=

P(A)

(5)

isdefinedtobetheconditionalprobabilityofthe eventB

under

thecondition

A.

From (5) itfollows immediatelythat