Math Bac Cours 11

7/27/2019 Math Bac Cours 11

1/10

- 1 -

Calcul des Probabilits

:I.:

: :Exprience alatoire.1:

.:19.( ):pile P( ):face F: .( )ventualitF.P = { },F P .( )Univers des ventualits ( )vnement lmentaire

{ }A P=. .( )vnement impossible .( )vnement certain

.:29 :{ }, , ,PP PF FP FF =.:( ) 4Card =. .{ }FP{ }FF{ }PF:{ }PP

:39 61 :{ }1,2,3,4,5,6 =. ( ):.{ }2{ }3{ }4{ }5{ }6:{ }1 6Card =.

:49 =CubiqueDs.61 :{ }1 1,2,3,4,5,6 =. :

( ){ }21 1 1, /x y x y = = .:( ) 36Card =.

7:A ): ) ( ) ( ) ( ) ( ) ( ){ }1,6 ; 2,5 ; 3,4 ; 4,3 ; 5,2 ; 6,1A ( ):.= 6Card A =.( ) 236 6Card = = A :7 AACA =:{ }/A A . .)complmentairevnement(:.2 .

.

BA =( )B A)contraires(vnements


2/10

- 2 -

.A:AA: .

.A AA:AA: = . .

II.:Espaces Probabiliss Finis ::.1

:.i .1.ii .

:.2

:.i = ): ) ( ) ( ) ( )p p p p = = + ( ):. 0p =..iiA B:( )B A B A= ( )A B A =

): ) ( ) ( )p B p A p B A= + :( ) ( ) ( ) 0p B p A p B A = ): ) ( )p A p B..iii:A A = A A = .:( ) ( ) ( ) ( )1 p p A A p A p A= = = +

( ): ( )1p A p A= .

AB :A B = . ( .( A et B sont incompatibles:AB

.

. :[ ]0,1( )Ppi.( ) 1p =.

ii.( ) ( ) ( )p A B p A p B= +

.( )PBA ,( )( ) ,p p .

,( )( ) ,p p .( ) [ ]

( )

: 0,1p

A p A

P( )pBA. :

i.( ) 0p =.

ii.( ) ( )A B p A p B

iii.( ) ( )1p A p A= .

iv.( ) ( ) ( ) ( )p A B p A p B p A B= +


3/10

- 3 -

.iv:( )A B A B A= ( )A B A = . ): ) ( ) ( )p A B p A p B A= + .): ) ( )B A B B A= ( ) ( )A B B A = ): ) ( ) ( ) ( ) ( ) ( )p B p A B p B A p B A p B p A B= + = ): ) ( ) ( ) ( )p A B p A p B p A B= + .

III.:Hypothse dEquiprobabilits :,( )( ) , p P.:{ }1 2, ,..., n =:( )Card n =( )*n .

:.1

:{ }( )1k p:.2 =.:{ }( ) { }( ) { }( ) { }( ) { }( ) { }( )1 2 3 4 5 6p p p p p p k = = = = = = }: } { } { } { } { } { }1 2 3 4 5 6 = + + + + +. ): ) { }( ) { }( ) { }( ) { }( ) { }( ) { }( )1 2 3 4 5 61 p p p p p p p = = + + + + +

( ){ }: { }( )( )1 1

1 11 n p p

n Card = = =

}: } { }( )( )

1 11,2,..., :

kk n p

n Card = =

:

.3 :

AA: .:{ }1 2, ,..., kA a a a= :( ) { }( ) { }( ) { }( )

( ) ( )( )( )1 2

1...

k

Card Akp A p a p a p a k

Card Card Card = + + + = = =

.

.

9 .19 .109 ( ) [ ]: 0,1p P:

( ).0 0p =.p 1. .p

A9 .A,( )( ) , p P

A :( )

( )( )

=Card A

P ACard


4/10

- 4 -

:.4 . .-.1 .

- . - .- .

.2 ..3 ..4 ..5 . .11:::A-.1 A. :

( )( )( )

2

2

2

11

1

55

Card A Cp A

Card C = = =

:B- B. :( )

( )

( )

2

4

211

6

55

Card B C

p B Card C = = =

:C- C. :( )

( )( )

2

2

2

11

1

55

Card C Cp C

Card C = = =

:D- D. :( )

( )( )

2

3

2

11

3

55

Card D Cp D

Card C = = =

:M.2 M. :( )

2 2 2 2

2 4 2 32

11

1 6 1 3 155 5

C C C C p MC

+ + + + + += = =.

=:M A B C D ABCD.:( ) ( ) ( ) ( ) ( )

1 6 1 3 1

55 55 55 55 5p M p A p B p C p D= + + + = + + + =

:N.3 N. NM :( ) ( ) ( ) 1 41 1

5 5p N p M p M= = = =

:.4 F F. :( )

2

5

2

11

10 255 5

Cp FC

= = =.

:G.5 G. :( )

1 1 2

3 8 3

2

11

3 8 3 27

55 55

C C Cp G

C

+ += = =.( )V V V V

IV.:La Probabilt Conditionnelle ::1.1 . .

::A:B

( ):.1( )

p A B

p A

.


5/10

- 5 -

.2 )( )Ap B(.AB.3 :.1

( ):. ( )( )

1 1

3 4

2

5

3 4 3

5 4 5

Card A A Ap A

Card A

= = = =

.

.: ::A B( ): ( )

( )

1 1

3 2

2

5

3 2 3

5 4 10

Card A B A Ap A B

Card A

= = = =

.

( ):( )

3

1103 2

5

p A B

p A= =

.

AB.2 ( ): 121

4

2 1

4 2A

Ap B

A= = =.

A . .U( ):.3 ( )

( )A

p A Bp B

p A=

.

:.2

:.3

250:2.4 :

.250 .

,( ) pA( )P( ) 0p A .AB ( ):

( )

p A B

p A

( )Ap B( )/p B A:( ) ( )

( )Ap A B

p Bp A

=

:9AB ( ): ( )

( )A

Card A Bp B

Card A=

A9 : ( ) [ ]( )

: 0,1A

A

p

B p B

P


6/10

- 6 -

:.1:G .: F.:E.:I.

.2 :( ):.1 ( )( ) 200 4250 5Card Gp G Card= = =.

( )( )( )

50 1

250 5

Card Fp F

Card= = =

.

( )( )( )

110 11

250 25

Card Ep E

Card= = =

.

( )( )( )

140 14

250 25

Card Ip I

Card= = =

.

.2 ( ): 80 2200 5

Gp E = =.

.5 : :Formule des Probabilits Composes:

( ):: ( )( )

( ) ( ) ( )A Ap A B

p B p A B p A p Bp A

= =

.

( ): ( )( )

( ) ( ) ( )B Bp A B

p A p A B p B p Ap B

= =

.

:

V.:Les Probabilits Totales :: :Partition dun ensemble.1 :{ }1.= { }1,2,3,4,5,6: 2,4,6A ={ }2 1A ={ }3 5,6A =. :1 2A A = 1 3A A = 2 3A A = 1 2 3A A A = .

1A

2A

3A . .3A2A1A

:.2

BA :( ) ( ) ( ) ( ) ( )A Bp A B p A p B p B p A= =

BA :( )( )

( )( )

B

A

p A p A

p B p B=

,( ) p*n . nA. . .2A1A :. -i j

A A = 1 i n 1 j n i j. -1 2

...nA A A =


7/10

- 7 -

:.3

::1

A2

A...n

A.B :B B=

( )1 2 ... nB B A A A=

( ) ( ) ( )1 2 ... nB B A B A B A= 1A2A...nA1B A2B A...

nB A.:( ) ( ) ( ) ( )1 2 ... np B p B A p B A p B A= + + + ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 21 2...

nA A n Ap B p A p B p A p B p A p B= + + +

:.4 : :

1U: .2U: . :

19 .1U29

2U.

.1 ..2 ..3 .1U .1U:.1:1:A:

2:A2.:2U.

1A

2A .

::B.1 .B :( ) ( ) ( ) ( ) ( )

1 21 2

2 2 2 3 430,41

3 5 6 7 105A Ap B p A p B p A p B= + = + = .

::R.2 .R :( ) ( ) ( ) ( ) ( )

1 21 2

2 3 2 4 620,59

3 5 6 7 105A A

p R p A p R p A p R= + = + = .

.31U( )1Bp A. ): ) ( ) ( ) ( )

11 1B Ap B p A p A p B = :

( )( ) ( )

( )11

1

2 24 105 283 5 0,65

43 15 43 43

105

A

B

p A p Bp A

p B

= = = =

:

,( ) p *n . B.BnA...2A1A :( ) ( ) ( ) ( ) ( ) ( ) ( )

1 21 2...

nA A n Ap B p A p B p A p B p A p B= + + +

1-1-1-1-2-2


8/10

- 8 -

VI.:Lindpendance ::.1

: ( ) 0p A )B:A ) ( )Ap B p B= .:.2 :.61

:A2.:B7.:C.

:( )p A B( )B:( )p A.1AB.

CA.2 :C.1 :( ) ( )

( )

1 1

2

1 6 6 1

6 36 6

Card Ap A

Card

= = = =

.

): ) ( ) ( ) ( ) ( ) ( ){ }1,6 ; 2,5 ; 3,4 ; 4,3 ; 5,2 ; 6,1B C= :( )

( )( ) 2

6 1

6 6

Card Bp B

Card= = =

.

:( ){ }2,5A B ( ):= ( )( )

1 1

2

1 1 1

6 36

Card A Bp A B

Card

= = =

.

):.2 ) ( ) ( )p A B p A p B= AB.C.3 :( ) ( )

( )

2

2

3 9 1

6 36 4

Card Cp C

Card= = = =

.

:( ){ }2,2A C A C= :( )

( )( )

1 1

2

1 1 1

6 36

Card A C p A C

Card

= = =

.

( ): 136p A C )= ) ( ) 1 1 16 4 24p A p C = = ): ) ( ) ( )p A C p A p C .AC.

,( ) p.AB :( ) ( ) ( )p A B p A p B=


9/10

- 9 -

: :Lindpendance des preuves alatoires.3 :.9(in) . )9(iin m ( )n m.

( )9(iiin m .9(ivn.

. .

): )i( )iii( )iv.) )ii.

... )(

. : :Les preuves rptes.:9

. . n .

.:9.3

: .: .3::S:1:{ }3,6S = ( )

6

1

3

2p S = =.

.S

.S:45CS: ( ) 1

3p S =E S= :

( ) ( ) ( )21

13

13

p E p S p S= = = =.

SESSS :( )( ) ( ) 4 14 1 23 3

p S p S

=

:.3 :

4 1

4

5

1 2 100,41

3 3 243C

=

:2 . 3:A A :


10/10

- 10 -

( )( )( )

4 14 14

5

4

5

4 1

4

6 5

2 4 2 1

3 36

24

6 6

Card Ap A C

CC

rd

C

a

= = = =

:

.:.1 2

3. .

..2 10 ..2 .1

VII.::1.1 B1 B2 :. B1 B2.

.1 :E:F:

:S.2( ):- 17

30p S =.

- . S:2.2 ( .) :

. .

.-.1- .

.2 .:3.3 10%20%.40%60% . .

.1 :.. ..

.2 ..3 .

,( ) p*n Sp. :(kSn

0 k n ( :( )1n kk k

nC p p

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Math Bac Cours 11