 . 3 
 


              .      ,         .       .       ,    ,     .





 

 . 3 





:

1.   .

2.   .

3.      .

4.      .

5.      .

6.     .

7.    .

8.      .

9.    ,     ;    .



    .

       ,        ;    .

    (, ).

   -          ().








         .

   ,          .

          .

   .

     probabilite ().

       A, B, C, D, , H

1.   .

       ,   .

       ;   ;   (          ).

      ( ).

 ()    ;          .

   .

         ,     ().

P (A) = m / n

P (A) -   .

n -   .

m -   .

   .

1.         (      ).

2.     .








3.     .

 (?) = 0

1.   , ,     n,       .

0 ? () ? 1

0 % ? () ? 100 %

    m    .

2.  n = n;

        ,     ;    .






,     (  )   .

3.  m = 0;

     ,     ;    .

P (?) = 0 / n = 0

,     (  )   .

 22. 8

              70.

  ,        ,    .

:

  3  , , .

30 , 31 , 31 .

     .

30 + 31 + 31 = 92 .

  ,   :

92 - 70 = 22 .

    .

 - ,   .

 () =  -  .

n -   .

m -   .

P (A) = M = m / n

P (A) = M = 22 / 92 = 11 / 46

: 11 / 46

 22. 9

     1 000,     .

    .

:

n = 1 000 -   ( ;  )

m = 5 -    ( ).

P (A) = M = m / n

P (A) = 5 / 1 000 = 0, 005

:  0, 005

  .

   .   ,   .

   ,    .

 - ,    .

 () = m / n.

n -  .

n = 5

   .

m -  .

m = 0

     .

P (A) = 0 / 5 = 0

         .

:  0.

    ;    .

 .

      .

 - ,     .

 () = m / n  ( ).

n = 5

   .

m -  .

m = 5

     M .

P (A) = 5 / 5

    ;    .

: 1

  .

      .

  1, 2, 3, 4, 5, 6 .

     .

 - ,    .

n -  .

n = 6

    ,       ;     .

n -  .

n = 3

    .

P (A) = 3 / 6 = 1 / 2 = 0, 5

: 0, 5

2.   .

   g     G.

   -         g.

    a    g   G.

P (A) = Sg / SG

 g    .

,     ,  :

1.  (, ).

2.   (  ).

3.  (  ).

   (,   ) - mes.

mes -   .

  .

                .

 () = mes g / mes G

 - ,   g.

n = mes g  -    .

m = mes G -  .

l -    .

S -  .

V -  .

,      ,    ,   .

P (A) = lg / lG -  

P (A) = Sg / SG -  

p (A) = Vg / VG -  .

3.      .

 1.

 500      15 .

 ,      .

:

n = 500

m = 500 - 15 = 485

:

P () = ?

:

 - ,    .

n = 500 ( ).

m = 500 - 15 = 485 ( ).

     ,  .

P (A) = n / m = 485 / 500 = 485 ? 2 / 500 ? 2 = 970 / 1 000 = 97 / 100 = 0, 97

: 0, 97

 23. 19.

    ,  :

1)  .

2)    .

 ,       .

:

     -   .

1)    ,  .

     10 .

 - ,     .

,    ,   .

n = 10

    10, :

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

m = 1

          .

P (A) = m / n

P (A) = 1 / 10

2) B - ,    .

 - ,    .

   B.

   ,      .       .

n = 9 (   )

m = 1 (     )

P (B) = 1 / 9

    .

   ,        .

n = 10 ( ).

m = 1 (    ).

 () = 1 / 10

  A  B ;           .

    ; ,   .

 ( ? ) = P () ? P (B)

       .

P (B) = 1 / 10 ? 1 / 9 = 1 / 90

: 1) 1 / 10; 2) 1 / 90

 20. 32.

          .

    ;      :

1) .

2)   .

:

1)  - ,        .

B - ,        .

   .

n = 5

     (1, 3, 5, 7, 9)

m = 1

       .

P (A) = m / n = 1 / 5

   .

n = 5 (    ).

m = 1 (      ).

P (B) = m / n = 1 / 5

 A  B    ; ,   .

 ( ? ) =  () ?  (B)

        .

P ( ? B) = 1 / 5 ? 1 / 5 = 1 / 25

: 1 / 25

2)  - ,    .

B - ,         .

    .

n = 5

       (2, 4, 6, 8, 0).

m = 1

      .

 () = m / n = 1 / 5

    .

n = 4

       .    ;     ().

m = 1

       .

P (B) = m / n = 1 / 4

  A  B    ; ,   .

 ( ? B) = P (A) ?  ()

 (A ? B) = 1 / 5 ? 1 / 4 = 1 / 20

: 1 / 20

 23. 22.

        (    ).

      5 / 6;       1 / 6.

        .

:

    6 .

n = 6 ( ).

       :

P (A) = 5 / 6 = m / n

m = 5 ( ).

    .

      :

P (B) = 1 / 6 = m / n

m = 1 ( ).

    .

:  5 ; 1 .

 23. 23.

        .

    ;   ,        2 / 7.

:

 x -    .

A - ,    .

 (A) = m / n

n = x + 4 ( ).

m = 4 ( ).

4 / (x + 4) = 2 / 7

2 (x + 4) = 4 ? 7

 + 4 = 4 ? 7 / 2

x + 4 = 14

x = 14 - 4

x = 10 ( ).

: 10 .

 23. 24

       .

  , :

1)          .

2)        ().

3)      9.

:

1)  .

 - ,           .

   .

    90.

n = 90 ( ).

   .

   .

10, 20, 30, 40, 50, 60, 70, 80, 90,

21, 31, 41, 51, 61, 71, 81, 91,

32, 42, 52, 62, 72, 82, 92,

43, 53, 63, 73, 83, 93,

54, 64, 74, 84, 94,

65, 75, 85, 95,

76, 86, 96,

87, 97, 98.

m = 45  ( ).

P (A) = 45 / 90 = 1 / 2

: 1 / 2

 .

      .

     n1 ;     n2 ,  n -     nk .

    .

     ;     ;      .

m = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 ().

P (A) = 45 / 90 = 1 / 2

: 1 / 2

2) B - ,        .

   .

    90.

n = 90 ( ).

   .

   .

11, 22, 33, 44, 55, 66, 77, 88, 99.

m = 9

P (B) = 9 / 90 = 1 / 10

:  1 / 10

3) C - ,      .

   .

n = 90 ( ).

   .

   .

    ,      18.

          ( 9).

18, 27, 36, 45, 54, 63, 72, 81, 90, 99

m = 10 ( ).

    .

n = 99 / 9 = 10

     ,   .

P (C) = 10 / 90 = 1 / 9

: 1 / 9

 23. 25

    1, 2, 3  .

  ,       .

:

 - ,       .

n = 3 ( ).

m = 2 (    ).

 () = 2 / 3

: 2 / 3

 23. 26.

         .

  ,    .

:

 - ,    .

n = 3

m = 2

P (A) = 2 / 3

:  2 / 3

 23. 27.

        .

     ,   ,          ,   1?

:

    -  1, 2, 3, 4, 5 .

   .

      .

P (A) = m / n

n = 5 + 7 = 12  (   ).

m = 7 ( ).

P (A) = 7 / 12

    .

        ,           .

:  8 .

 23. 28.

    ;  ;   .

      ,   ,          ,   .

:

     -  ? 1.

       ,     .

,        ,         .

n = 7 + 2 + 11 = 19 ().

: 19 .

 23. 29.

   .

 ,  :

1)  .

2)   .

3) ,    .

4) ,    .

5) ,    .

:

    ,      -    .

        .

   1, 2, 3, 4, 5, 6

  1, 2, 3, 4, 5, 6

     ,         .

           .

,   ,   .

          .








1) 1 - ,    .

m = 1 ( ).

 1  1    .

P (A1) = m / n = 1 / 36

:  1 / 36

2) 2 - ,     .

1 1, 2 2, 3 3, 4 4, 5 5, 6 6

m = 6 ( ).

P (A2) = m / n

P (A2) = 6 / 36 = 1 / 6

:  1 / 6

3) 3 - ,   ,    7.

1 + 6; 6 + 1; 2 + 5; 5 + 2; 4 + 3; 3 + 4

m = 6 ( ).

P (A3) = m / n

P (A3) = 6 / 36 = 1 / 6

: 1 / 6

4) A4 - ,   ,    10.

  ,   -  > 10.

5 + 6; 6 + 5; 6 + 6

m = 3 ( ).

P (A4) = m / n

P (A4) = 3 / 36 = 1 / 12

:  1 / 12

5) 5 - ,   ,    .

2 ? 3; 3 ? 2; 1 ? 6; 6 ? 1

m = 4 ( ).

P (A5) = m / n

P (A5) = 4 / 36 = 1 / 9

: 1 / 9

 23. 30.

     .

  , :

1)       ;        .

2)      ,    .

3)       .

:

n = 6 ? = 36








1) 2 - ,        ;       .

    < 4.

    > 4.

1 5; 1 6; 2 5; 2 6; 3 5; 3 6

m = 6 ( ).

 (1) = m / n

P (A1) = 6 / 36 = 1 / 6

: 1 / 6

2) 2 - ,          .

1 6; 1 5; 1 4; 1 3; 1 2;

2 6; 2 5; 2 4; 2 3;

3 6; 3 5; 3 4;

4 6; 4 5;

5 6.

m = 15

P (A2) = m / n

P (A2) = 15 / 36 = 5 / 12

: 5 / 12

3) A3 - ,         .

1 + 4; 4 + 1; 2 + 3; 3 + 2

P (A3) = m / n

P (A3) = 4 / 36 = 1 / 9

: 1 / 9

 23. 31

 ,       :

1)     ,   ;     .

2)      .

3)      ,    .

:

     ,        ,     .

  ( )   .

,   ,   .

        (     );    (   ).






1) 1 - ,      ,   ;     .

1   < 5;

2   > 4

1 5; 1 6; 2 5; 2 6;

3 5; 3 6;

4 5; 4 6

m = 8 ( ).

P (A1) = m / n

P (A1) = 8 / 36 = 2 / 9

:  2 / 9

2) 2 - ,       .

1 6; 2 6; 3 6; 4 6; 5 6; 6 6.

m = 6 ( ).

P (A2) = 6 / 36 = 1 / 6

: 1 / 6

3)      ,    .

6 5; 6 4; 6 3; 6 2; 6 1;

5 4; 5 3; 5 2; 5 1;

4 3; 4 2; 4 1;

3 2; 3 1; 2 1

m = 15 ( ).

P (A3) = m / n

P (A3) = 15 / 36 = 15 / 12

: 15 / 12

 23. 32

  ϸ      .

     ,   ;       ,   ϸ.

         ?

:








      ,     .

1 + 5; 5 + 1; 2 + 4; 4 + 2; 3 + 3.

m = 5

     ,     .

1 + 6; 6 + 1; 2 + 5; 5 + 2; 3 + 4; 4 + 3

m = 6

A - ,   ;    .

 - ,   ϸ;    .

P (A) = 5 / 36

P (B) = 6 / 36

       .

:   ϸ.

4.      .

 84.

  [- 1, 2; 3, 8]       .

 ,     :

1) .

2) .

:

[- 1, 2; 3, 8] - .

)  -  .

)  -  .

:

P (A) - ?

P (B) - ?

:

P (A) = m / n

)  - ,   .

   .

   -  lG.

     .

n = lG = 3, 8 - (- 1, 2) = 3, 8 + 1, 2 = 5

   .

   ,     lg -   .

m = lg = 0 - (- 1, 2) = 1, 2

P (A) = lg / lG = 1, 2 / 5 = 12 / 50 = 24 / 100 = 0, 24

:  0, 24

) B - ,   .

  .

n = lG = 3, 8 - (- 1, 2) = 3, 8 + 1, 2 = 5

   .

   ,     lg -   .

m = lg = 3, 8 - 0 = 3, 8

P (B) = m / n

P (B) = 3, 8 / 5 = 38 / 50 = 76 / 100 = 0, 76

     )   .

    B .

      .

P (A) + P (B) = 1

P (B) = 1 - P (A)

P (B) = 1 - 0, 24 = 0, 76

: 0, 76

 85.

  [- 6; 14]       .

 ,       - 2,   5.

:

[- 6; 14] - .

  ? - 2 < x < 5

:

P (A) - ?

:

P (A) = m / n

   .

       .

n = lG = 14 - (- 6) = 14 + 6 = 20

   .

        ,     .

m = lg = 5 - (- 2) = 7

P (A) = 7 / 20 = 35 / 100 = 0, 35

: 0, 35

 86.

  1    .

  ,       80 .

    8  -  ? 80.

   ,    ,  80 .

,      ,  20 .

       ,    20   ,  ,   .

    20   .

 - ,    .

B - ,    .

P (A) = m / n = lg / lG

lG = n = 1  = 100 . -  .

lg = m = 20 . -  .

P (A) = 20 / 100 = 1 / 5

P (B) = m / n = lg / lG

lG = n = 1  = 100 .

lg = m = 20 .

P (B) = 20 / 100 = 1 / 5

 A  B ;         .

P (A + B) = P (A) + P (B)

          .

P (A + B) = 1 / 5 + 1 / 5 = 2 / 5 = 4 / 10 = 0, 4

: 0, 4

 88.

   R   .

  ,          .

:

  R.

R -   .

  ABC   .

:

P (A) = ?

:








     ,       .

   ,   .

   ,     .




  .


   .

   ,     (https://www.litres.ru/book/vasilisa-volkonskaya/teoriia-veroiatnostei-3-chast-74709278/)  .

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