math bac cours 11
TRANSCRIPT
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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
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.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= +
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.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
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:.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
.
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.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
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:.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 =
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:.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
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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=
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: :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 :
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( )( )( )
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