Foundations of the theory of probability

§

5. Independence 11

1

=

1) fortheindependenceoftwoeventsA

x

andA

2

:

?UiA 2 )

=P(A

1

)P(A

2

). (5)

Thesystemofequations

(2)

reducesitself

}

inthiscase,tothree

equations,besides

(5)

:

PiAiAz)

=

P(A

1

)P(A

2 )

?{A

X

A

2 )

=P(A

1

)P(A

a

)

?{A

X

A

2 )

=P(A

1

)P(A

2 ) ,

which obviouslyfollow from

(5).

11

Itneed hardlyberemarked thatfrom theindependence of

theeventsA

lt

A

2

,

..

.

,A

n

inpairs,i.e.fromtherelations

P(A«A,) =P(A

i

)P(A

i

)

«*>

itdoesnotatallfollowthat whenn>2 theseevents are

inde-

pendent

12

. (Forthatweneedtheexistenceofallequations (4).)

Inintroducingtheconceptofindependence,nousewasmade

ofconditionalprobability.Ouraimhasbeentoexplainasclearly

aspossible,in

a

purely

mathematicalmanner,themeaning

ofthis

concept. Its applications,

however,

generally

depend upon

the

propertiesofcertainconditionalprobabilities.

IfweassumethatallprobabilitiesP(A

g

(t

>) are

positive,then

fromtheequations

(3)

itfollows

13

that

P«>

...

4;;«>

MM

=P(4?)

. (6)

Fromthefactthatformulas

(6)

hold,andfromtheMultiplica-

tionTheorem (Formula

(7), §4),

follow

the

formulas

(2).We

obtain,therefore,

Theorem

II: A necessaryandsufficientcondition for

inde-

pendence

of

experiments5l

(1)

,

5l

(2)

,

...

,

9l

(w)

inthecase

of

posi-

11

P{4iZj

• P(A

X

)

• P{A

t

A

2

)

a*

P{A

X

)

• P(A^9{A

%

)

=

P(^){t

• P(^
  2 )}

»P(4
1

)P(i"

a

)

,etc.

12

Thiscanbeshownbythefollowingsimpleexample(S.N.Bernstein)

:

LetsetEbecomposedoffourelements

J

1

,£

2

,£

3

,

<£,

;thecorrespondingelemen-

taryprobabilities

p

it

p

2

,p

3

,p

4

areeachassumedtobe

X

A

and

A

={^,^}r

JB-Wj.W. C'^ft.W,

Itiseasytocomputethat

P(A)

=

P(B)=P(C)

="%,

P(AB)=P(BC)

-P(AC)

=

%

=

(V 2

)

2

,

P(A£C)=.14

*(V 2

)

3

.

u

Toproveit,onemustkeepinmind

thedefinitionofconditionalproba-

bility(Formula

(5),§4)andsubstitutefortheprobabilitiesofproductsthe

productsof

probabilitiesaccordingtoformula

(3).