Foundations of the theory of probability

§

4. ImmediateCorollariesoftheAxioms 7

P(AB)=P(A)P

A

(B). (6)

And by

inductionweobtain

thegeneralformula (the

Multi-

plication

Theorem)

P(A

1

A

2

...A

n)

=

P(A

l

)P

Al

(A

2

)P

AlA

AA

3

)...P

Al

A

2

...A

n

• l

(A

n).

(7)

Thefollowingtheoremsfolloweasily

:

P

4

(5)g0,

(8)

P

A

(E)

=

1,

(9)

PAB+C)=?AB)+?AC).

(10)

Comparingformulae (8)

—

(10)

withaxiomsIII—

V,

wefindthat

thesystem
$

ofsetstogetherwiththesetfunctionP

A

(B) (pro-

videdAisafixedset),formafieldofprobabilityandtherefore,

alltheabovegeneraltheoremsconcerningP(B) holdtruefor

the

conditionalprobability P

A

(B) (provided theevent A is fixed).

Itisalsoeasytoseethat

P^(A)=1. (11)

From (6) andtheanalogousformula

P (AB)=P(B)P

B

(A)

weobtaintheimportantformula

:

PB{A)

=

^m,

(12)

whichcontains, inessence,theTheoremofBayes.

TheTheorem

on

TotalProbability:

Let

A

1

+A

2

+.

.

.+

A

n

—

E(thisassumesthattheeventsA

lf

A

2J

..

.

,A

n

aremutually

exclusive)andletXbearbitrary.Then

P(X)=

PiAJ

P

Al

(X)

+

P(A

2

)

P

At

(X)

+

...

+

P(A

n

)

P

An

(X).-

(13)

Proof

:

X

=

AiX+A

2

X+

.

..+A„X;

using

(4) wehave

P(X)=P(A

1

X)+P(A

2

X)+...+P(A„X)

andaccordingto

(6)

wehaveatthesametime

P(A

i

X)=P(A

i

)P

At

(X).

TheTheorem of Bayes: Let

A

1

+

A

2

+.

..

+

A

n

=

E and

X

bearbitrary,then

p (A

,

PWP^X)

x(

*

PiAJP^W

+

P(A

2

)P

A

,(X)

+ +

P(A

n

)P

A

„(X)'

(

>

i=

1,2,3,....,».