2.Πιθανότητες - Μαθηματικά γενικής παιδείας

Click here to load reader

  • date post

    29-Jul-2015
  • Category

    Documents

  • view

    356
  • download

    2

Embed Size (px)

Transcript of 2.Πιθανότητες - Μαθηματικά γενικής παιδείας

.

.

.

( ) . . , . 1, 2, 3,, , = {1 , 2 , 3 ,, } . . . . ,

.

, . . ().

2

.

,

, . ( ) = , . ,

() = 0. ,

Venn. , () .

-

1)

. ,

AB

. :A B

: = {x / x x }

: : 1) 2) 3) = .

3

.

2)

. , . :A B

A

B

: = {x / x x } : , : 1) 2) 3) =

3) .

, . : : = { x / x } : : 1) = 4) =

A

2) = 5) ( ) = 4

3) = 6) = = =

.

4) . . : : = {x / x x } : 1) = 5) A BA-B

(

) :

2) = 6)

3) = 7)

4) = 8)

: , :

= = ( ) = = =

, = .

: : 1) 2) 3)

4) 5

.

= {1 , 2 , 3 , , }.

{ 1 } , { 2 } , { 3 } , , { } 1 , 2 , 3 , , , { 1 } , { 2 } , { 3 } , , { } :

f 1 = 1 , f 2 = 2 , f 3 = 3 ,..., f =

.

: 0 f i 1 , i =1 , 2 , 3 ,..., ( 0 i ) f 1 + f 2 + f 3 + ....f =

1 + 2 + 3 + .... = =1

, ( ).

, .

( ) :() =

() = ()6

.

: 1) () = 2) () =() =1 () () 0 = =0 ()

3) 0 ( ) 1 , 0 () () . : = {1 , 2 , 3 ,, } .

= {1 , 2 , 3 ,, } . { i } , i = 1, 2, 3,..., , (i ), : 0 ( i ) 1 , i = 1, 2, 3,..., () = ( 1 ) + ( 2 ) + ( 3 ) +...+ ( ) =1

( i ) { i }. () = {1 , 2 , 3 ,., } ( ) = ( 1 ) + ( 2 ) + ( 3 ) + + ( ) ,

( )= 0 . :1 , i = 1, 2 , 3,..., = {1 , 2 , 3 ,, } , .

( i ) =

, ( ). :

( i ) = lim f i = lim

i , i = 1, 2 , 3,..., . 7

.

1 : :

(U) = () + () ( ) = ( ) = . ( ) = + ,

(U) = () + (). :

(U) =

(U) ()

=

() + () ()

=

() ()

+

() ()

=

= () + ( )

:

, , , , :( U U ) = ( ) + ( ) + ( ) .

2 :

( ) = 1 ()

= (1) = (2). (1), :( ) = ( ) + ( ) () = ( ) + ( ) (2)

1 = ( ) + ( ) ( ) = 1 ( ).

8

.

3 : : (U) = () + () ( )

: (U ) = () + ( ) ( ) . :

(1) ,

() + ()

(U ) =

(U ) ( ) () () +

=

() + ( ) ( ) ( ) ( ) ()

=

=

() ()

= () + () ( )

4 , . , ( ) ( ) . , ( ) ( ) . : ( ) ( ) ( ) ( ) ( ) ( ). ( ) ( )

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

() ( ) , ( ) ( ) , 9

() ( ) ,

) ( ) ( ) ,

.

5 :

( ) = () ( ): ( ) ( ) = (1) ( ) ( ) = (2).

(1), : ( ( ) ( ) ) = ( ) + ( ) (2)

( ) = ( ) + ( ) ( ) = ( ) ( )

: :

( ) = () ( )

6 :

( ( ) ( ) ) = () + () 2( ) , : ( ( ) ( ) ) = ( ) + ( ) = = ( ) ( ) + ( ) ( ) = = ( ) + ( ) 2 ( ).

( ) ( )

: , , ( ) ( ) ( ) = , :

( ( ) ( ) ( )) = ( ) +( ) + ( ) ( ) = ( ) + ( ) + ( )

10

.

Venn

() = 1

0 ( ) 1

() = 1

( ) = 0

: ( )

( ) = 1 ( )

( )

=

() = ( ) = = () ( )

11

.

( )

() = =() +() ()

( )

( ) = =() +() ()

( )

( ) ( ) ( ) ( )

( ) ( ) = =( ) +( ) = =()+()2( )

( )

= ( )1 De Morgan

( ) =[ ()] = =1()

( )

= ( )2 De Morgan

( ) = [()] = =1()

, ( )

() =() +() +() () () ( ) +()

12

.

1. : . . ) . ) : : . : . ) : , . 1 2

3

4

5

) ,

= {, , , , , , , , , }.) = { , } = { , , , , }.

) = { } = { } = { , , , , }.

13

.

2. - . : : - 1 : - 2 : - 3 : : . : ) ) To . ) ( ) ) ( ) ) ( ) .

To 3 . 3 .) To .

1 2 , ( ) 3 . ( ) 3 .) = ,

2 3 . o ( ) = = .) To .

1 2 . ( ) 3 .

14

.

3. : : 4 : . , : ) ) ) )

1

2 2 2

3 3 3

4 4 4

5 5 5

6 6 6

1 1

. :

= { 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6 } () =12. , .) : = { 1, 2, 3, 1, 2, 3 } ( ) = 6, () =

( ) 6 1 = = . () 12 2

) : = { 1, 3, 5 } () = 3, () =

() 3 1 = = . () 12 4 ( ) 2 1 = = . () 12 6( ) 7 = . () 12

) : = { 1, 3 } ( ) = 2, ( ) =

) :

= { 1, 2, 3, 1, 2, 3, 5} ( ) = 7, ( ) =:

( ) : ( ) = ( ) + () ( ) =

6 3 2 7 + = . 12 12 12 1215

.

4. = {1 , 2 , 3 , 4 , 5 , ,1000 } : : 4 : 6 . . : ) , .

)

)

)

)

) ={ 4, 8, 12, 16,...,1000}. 1 = 4, = 4 =1000. = 1 + ( 1) , 1000 = 4 + 4( 1) 4 = 1000 = 250. ( ) = 250, ( ) = ( ) 250 = = 0, 25 . ( ) 1000

) = { 6, 12, 18, 24,...,996}. 1 = 6, = 6 = 996. = 1 + ( 1) , 996 = 6 + 6 ( 1) 6 = 996 = 166. ( ) 166 () = 166, ( ) = = = 0,166 . ( ) 1000

) , 4 6, 12. = { 12, 24, 36,...,996}.

1 =12 = 12 = 996. = 1 + ( 1) , 996 = 12 + 12( 1) 12 = 996 = 83. ( ) = 83, ( ) = ( ) 83 = = 0,083 . 1000 ( )

250 166 83 333 + = = 0,333. 1000 1000 1000 1000 250 83 167 ) ( ) = ( ) ( ) = = = 0,167 . 1000 1000 1000

) ( ) = ( ) + ( ) ( ) =

16

.

5. 2 4 x . : ) 3 5 ) 3 5 ) 3 ) 3 5 ) 3 5 ) 3 5 . : = {240, 241, 242, 243, 244, 245, 246, 247, 248, 249}

: : 3 : 5 . , . : = { 240, 243, 246, 249} ( ) = 4, ( ) =

( ) 4 = = 0,4 . () 10

: = { 240, 245 } () = 2, () =

() 2 = = 0,2. () 10 ( ) 1 = = 0,1. 10 ( )

)

: = { 240 } ( ) = 1, ( ) =

) ) ) )

( ) = ( ) + ( ) ( ) = 0,4 + 0,2 0,1 = 0,5. ( ) = ( ) ( ) = 0, 4 0,1 = 0,3. ( ( ) ( ) ) = () + () 2( ) =0, 4 + 0,2 2 0,1 = 0, 4. ( ) = ( ( ) ) = 1 ( ) =1 0,5 = 0,5.

) ( ( ) ) = 1 ( ) =1 0,1 = 0,9.

17

.

6. = { 1 , 2 , 3 , 4 } . ) ( 1 ) =1 1 1 , ( 2 ) = , ( 3 ) = , ( 4 ) . 2 4 81 ( 2 ) = 3 ( 3 ) , ( 2 ) ( 3 ) . 6

) ( 1 ) = ( 4 ) =

) = { 1 , 2 } , = { 2 , 4 } , ( 1 ) = (3 ). ) :

1 1 7 ( ) = , , ( ) = 4 2 12

( 1 ) + ( 2 ) + ( 3 ) + ( 4 ) = 1 ( 4 ) = 1 ( 1 ) ( 2 ) ( 3 )

1 1 1 1 ( 4 ) = 1 ( 4 ) = . 2 4 8 8

) : ( 1 ) + ( 2 ) + ( 3 ) + ( 4 ) = 1

16

+ 3 (3 ) + (3 ) +

16

=1

1 1 2 1 4 ( 3 ) = 1 4 ( 3 ) = ( 3 ) = .6 6 3 6

( 2 ) = 3 (3 ) = 3

16

=

12

.

) : ( ) = ( 1 ) + ( 2 ) ( ) = ( 2 ) + ( 4 )

1 1 1 1 1 = + (2 ) (2 ) = (2 ) = . 2 4 2 4 4

7 1 7 1 1 = + ( 4 ) ( 4 ) = ( 4 ) = . 12 4 12 4 3

( 1 ) + ( 2 ) + ( 3 ) + ( 4 ) = 1 ( 3 ) = 1 ( 1 ) ( 2 ) ( 4 )

1 1 1 1 ( 3 ) = 1 ( 3 ) = . 4 4 3 6

18

.

7. = {0 , 1 , 2 , 3 , 4 , , 50 }. ( ) =

1 , 3

= 1, 2, 3, 4 , ..., 50 , :) ( 0 ) ) ( ) = { 2 , 4 , 6 , 8 , , 50 }.

) : ( 0 ) + (1) + ( 2 ) + ( 3) + ( 4 ) + ... + ( 50 ) = 1 .

() =

1 , = 1, 2 , 3, 4 ,....,50 : 3

1 1 1 1 1 1 1 1 1 1 (0) + + 2 + 3 + 4 + ... + 50 = 1 (0) = 1 + 2 + 3 + 4 + ... + 50 3 3 3 3 3 3 3 3 3 3

(1).

1 1 1 1 1 + 2 + 3 + 4 + ... + 5 0 3 3 3 3 3

50 1 1 = . 3 350

1 =

1 1 1 1 350 1 50 50 3 3 1 ( 1) = 3 3 = 3 1 , S = S50 = 1 2 2 350 1 1 3 3

(1) (0) = 1

350 1 350 + 1 (0) = . 2 350 2 350

) : ( ) = ( 2 ) + ( 4 ) + ( 6 ) ++ ( 50 ) =

1 1 1 1 + 4 + 6 + ..... + 50 , 2 3 3 3 3 1 1 = 2 , 2 3 3

25 1 =25

1 1 1 1 350 1 2 2 2 50 50 3 3 = 3 3 = 3 1 . () = S25 = 1 8 8 350 1 2 32 3

19

.

8. = { 1 , 2 , 3 , 4 } . 1 1 , , , 1 , 16 4 2 4 { 1 } , { 2 } ,

{ 3 } { 4 }. 1 1 ( 4 ) = ( 1 ) = 1 , ( 2 ) = , ( 3 ) = : 4 2 4 16 ( 1 ) + ( 2 ) + ( 3 ) + ( 4 ) = 1

0 ( i ) 1 i = 1, 2 , 3 , 4.

: ( 1 ) + ( 2 ) + ( 3 ) + ( 4 ) = 1 1 +16 1 1 + + =1 42 +8+4+2 = 0 4 2 4 16

3 2 12 = 0 2 = 4 = 2 = 2 . = 2 :

( 1 ) = 1

2 1 1 1 1 1 2 1 = , (2 ) = = , ( 3 ) = = ( 4 ) = = , 4 2 22 4 42 8 16 8

= 2 . = 2 :

(2 ) =

1 1 = [ 0 , 1 ] , = 2 . 2 ( 2) 4

20

.

9. 5 , x y . . 3 3 , , : 4 5

) x , y . ) . ( ) = x + y + 5 . : : , ( ) = 5 , : , ( ) = x : , ( ) = y . () =() 5 () x y () = = , () = = . () x + y + 5 () x + y + 5 x + y+5

) 3 ( ) = . , : 4 3 x y 3 x+y (U) = () + () = + = 4 x + y+5 x + y+5 4 x + y+5

3x + 3y +15 = 4x + 4y x + y = 15 (1) . 3 ( ) = . , : 5 3 5 y 3 5+ y (U) = () + () = + = 5 x + y+5 x + y+5 5 x + y+53x + 3y +15 = 25 + 5y 3x 2 y = 10 (2) .

(1) (2) x = 8 y = 7, 5 + 8 + 7 = 20 . ) , : ( ) = ( ) + ( ) =

( ) ( ) 5 8 13 + = + = . () () 20 20 2021

.

10. 55% , 40% 30% . . : ) ) ) ) ( 2003) : : : () = 0,55 () = 0,40. 30 % ( ) = 0,30.

: ( ) = 1 () = 1 0,55 = 0, 45 .

:

) : ( ) = ( ) + ( ) ( ) = 0,55 + 0, 40 0,30 = 0,65 , 65%.

) : ( ) = ( ) = ( ) ( ) = 0,55 0,30 = 0, 25 , 25%.

) : ( ) = ( ) = ( ) ( ) = 0, 40 0, 30 = 0,10 , 10%.

) : ( ) = () + ( ) ( ) = 0, 45 + 0, 40 0,30 = 0,75 , 75%.

22

.

11. 7 10

,

17 20

,

: ) . ) ().

2 5

, ()

( ) ( ) , : ( ( ) ( ) ) = 7 10 ( ) + ( ) 2 ( ) = 7 10

(1).

, : ( ) = 17 20 ( ) + ( ) ( ) = 17 20

(2).17 7 3 20

) (2) (1) ( ) =

20 10

=

(3).

) , : ( ) = 2 5 ( ) ( ) =(3)

2 5

( )

3 20

=

8 20

( ) =

11 20

(4).

(2) : ( ) + ( ) ( ) = () = 9 2023

17 20

(3),(4)

11 20

+ ( )

3 20

=

17 20

( ) =

17 20

+

3 20

11 20

.

12. , , ( ) = ( ) ( ) . ) ( ) =1 6 , ( ) = 2 3

( ) > ( ) , :

i) ( ) ( ) . ii) ( ) . ) ( ) = 1 , ( ) = 1 ( ) = 1 . ( ) =1 6 ( ) ( ) = 1 6

(1).2 3 = ( ) + ( ) 1 6

) i) ( ) = ( ) + ( ) ( )

( ) + () =

5 6

(2).

(1) (2) () () 5 1 1 1 x 2 x + = 0 . x = x = . 6 6 3 2

( ) > ( ) , ( ) =

1

1 ( ) = . 3 2 1 2 1 6 = 1 3 .

ii) ( ) = ( ) = ( ) ( ) =

) ( ) = ( ) + ( ) ( ) 1 = ( ) + ( ) ( ) ( ) ( ) 1 + ( ) ( ) = 0 ( ( ) 1) ( ) ( ( ) 1) = 0

( ( ) 1) ( ( ) 1) = 0 ( ) 1 = 0 ( ) 1 = 0

() =1 () =1.

24

.

13. . ( ) = 60 , , 25 ( ) + 9 ( ) = 20 ( ) + 6 ( ) 5 2 2

(1) , :

) () (). ) . ) ( ). ) (1) :1 3

,

( 25 ( )

2

20 ( ) + 4 + 9 ( ) 6 ( ) + 1 = 0 2

) (

)

(5 ( ) 2)

2

2 () = 5 () 2 = 0 2 5 + ( 3 ( ) 1) = 0 3 ( ) 1 = 0 ( ) = 1 3

) , . ( ) = ( ) ( ) ( ) = ( ) ( ) ( ) = 2 5 60 ( ) = 24

() =

1 ( ) = ( ) ( ) ( ) = 60 ( ) = 20. () 3

()

24 20 .)

( ) ( ) , : (( ) ( )) = 1 3 2 1 1 1 + 2 ( ) = ( ) = . 5 3 3 525

( ) + () 2 ( ) =

1 3

.

14. , P( < P() ) ( ) ( ) 8 2 6 + 1 = 0 , :

) ( ) ( ) . ) . i) . ii) . ) 82 6 +1 = 0 =(6)2 481 = 4 > 0, 62 1 1 = = . = 12 4 2 1 1 ( ) < ( ) , ( ) = ( ) = . 4 2 ) 1 4

,

= ( ) , :

((

) =

)

1 4

1 ( ) =3 4 1 4 + 1 2

1 4

( ) = 3 4

3 4

( ) + ( ) ( ) =

( ) =

( ) = 0.

.i) ,

:

1 ( ) = ( ) ( ) = ( ) = . 4

ii) ( ). . =, ( ) = = , ( ) = ( ) = 1 .

26

.

15. , ( ) ( ) x , 5 f ( x) = 2x3 + x2 x +1 ( ), 2

: ) () (). ) ( ) , ( ) ( ) . ) x R f ( x ) = 6x 2 + 5x 1 .

f ( x ) = 0 6x 2 + 5x 1 = 0 6x 2 5x + 1 = 0 x =f ( x ) > 0 6x 2 + 5x 1 > 0 6x 2 5x + 1 < 0

1 3 1 3

x= 1

. , .) , ( ) ( ) ( ) ( )

2 5

(1).

3 20

(2). (2)

: ( ) + ( ) ( )

3 20

2 5

+

3 4

( )

3 20

( )

2 5

+

3 4

3 20

( )

8 + 15 3 20

( ) 1,

, (2). (1) (2) ) = , :( ) () ( ) 1 () ( ) 1

3 20

( )

2 5

.

3

4

1 ( ) .

4

28

.

: 1) : ) : ( ) () ) , . 2) , . . , . 17. 4 [()]2 () + () 4 () () < 0 (1) . :

) () () ) 0 < () < 1 4

1 4 .

( ) 0 (2).

(1) : ( )[4 ( ) 1] ()[4 ( ) 1] < 0 ( 2) 1 [4 ( ) 1] [ ( ) ()] < 0 4 ( ) 1 < 0 ( ) < (3). 4 >0 ( ) 0 , , ( ) > 0 , ( ) 0 .

() = 0 , (2) () < 0 . 1 ( ) > 0 (4), (3) (4) 0 < () < . 4 1 1 ( ) (), ( ) < , ( ) < . 4 4

29

.

18.

, . () 0,28 () 0,71. : ) ( ) 1,01 ( ) ) . ( 1994) ) : ( ) 0, 28 1 ( ) 0, 28 ( ) 1 0, 28 ( ) 0,72

(1) (2). (3).

() 0,71 1 () 0,71 () 1 0,71 () 0, 29 (1) (2) () + () 1,01

: ( ) 1,01 ( ) ( ) 1,01 [ ( ) + () ( )] ( ) 1,01 ( ) () + ( ) ( ) + () 1,01

(3).) = , ( ) = 0 ,

( ) 1,01 ( ) 0 1,01 ( ) ( ) 1,01, 0 ( ) 1, .19. (), ( ) = 4 2 + 12 + 10 , R.

) R. ) () = 1.

4 2 + 12 +10 0 (1) ) 0 ( ) 1 0 4 +12 + 10 1 2 4 +12 + 10 1 (2) 2

(1) 2 2 + 6 + 5 0 R , = 4 < 0 = 2 > 0 . 3 2 (2) 4 2 +12 + 9 0 ( 2 + 3) 0 2 + 3 = 0 = . 2

) () :9 3 3 ( ) = 4 + 12 + 10 = 4 18 + 10 = 9 18 + 10 = 1. 4 2 2 2

30

.

20. , . () () :2 ( ) 5 + 3 + ( ) ( ) = ( ) + ( ) + 3

(1) ,

2 3

.

: 0 ( ) 1 (2) , : (2) 3 ( ) + 3 4 , ( ) + 3 > 0 , ( ) + 3 = ( ) + 3. (2) 0 2( ) 2 5 2( ) 5 3, 2( ) 5 0 16x 8 > 0 x > 1 2 1 2

xf

0

1 2

1+

01

f

x = () =

1 2

f ,2

.

1 1 1 f = 8 8 +1 = 1. 2 2 2

31

.

1. : : 4 : , :

)

)

)

.

2. = { 0, 2, 3} = { 2, 3, 5} .

) . ) , : 2 : x + x + 1 = 0 : x y = 2 y = x : x + x + x 6 = 0 1.3 2

03. = { 1 , 2 , 3 , 4 }. P (1 ) = P ( ) , = { 1 , 3 }.

1 1 1 P ( 2 ) = P ( 3 ) = P ( 4 ) , 2 3 6

04. = { 1 , 2 , 3 , 4 }. = { 2 , 3 } , = { 2 , 4 } P ( ) =1 2 1 , P () = P ( 2 ) = , P (1 ). 3 2 3

5. = { 1 , 2 , 3 , 4 } . ) ( 1 ) = ( 4 )1 3 ( 3 ) . , ( 2 ) = 5 12 ( 4 ) = 2 ( 3 ) ,

) ( 1 ) , ( 2 ) , ( 3 ) ( 4 ) 2 = { 1 , 2 } = { 3 , 4 } ( ) = ( ) + , 9 ( 1 ) , ( 2 ) , ( 3 ) ( 4 ) .

6) = { 1 , 2 , 3 } . ( 1 ) , ( 2 ) , ( 3 )

( 3 ) = 2 ( 1 ) . ( 1 ) , ( 2 ) ( 3 ) .

32

.

7. = { 0 , 1, 2 , 3 , 4 ,..., 30 } . ( + 1) = ( ) + (1) = 1 , 2 , 3 , ... , 29 (1) = 1 600

,

) (0). ) = { 0 , 2 , 4 , 6 ,..., 30 } , (). 8. = { 1 , 2 , 3 , 4 } . R , 1 6 , 1 2 , 1 12 , 12

{ 1} ,

{ 2 } , { 3} { 4 }.

= { 1 , 2 , 3 }.

9. , , ( ) = ( ) =

4 10

2 , ( ) = 10

8 . : 10) ) .

)

10. , , ( ) = 0, 2 ( ) = 0,6 . ) ( ). ) 0,1 .

11. ( ) () 18x 9x + 1 = 0 ( ) < () . ) ( ) () . ) .1 2

2

,

) ( ) . 0 12. P ( ) , = 1, 2, 3 , 4 , 5 , 6 3 . P (2 ) = P (4 ) = P (6 ) P (1) = P (3) = P (5) = P (2) , : 2 ) P ( ) , = 1, 2, 3 , 4 , 5 , 6 .

) : 3 . ) : 3 .

33

.

0 13. = { 1 , 2 , 3 , 4 }. P (1 ) =P ( 3 ) = 1 , P ( 4 ) = , , R . 15 30

2 1 3 6 , P ( 2 ) = , 3 5

) , . ) = { 2 , 3 , 4 }. 0 14. A , , P ( ) =1 , : 12 ) . P ( ) = 1 2 , P () = 5 3

) , .) , .

) . ) , . ) , . 0 15. A , , P ( ) = P ( ) = 1 , : 4 2 3 , P ( ) = 3 4

) P ( )

) P ()

) P ( )

) P ( )

0 16. , , . 1 1 1 = , P ( ) = 2P ( ) + P ( ) P ( ) = P ( ) + P ( ) 3 2 4 : ) P ( ) ) P () ) P ( )

0 17. , , . = P ( ) + P ( ) = 2 P ( ) P (). 0 18. , . P ( ) = P ( ) P ( ) = P ( ) , P ( ) = P ().

19. , , P ( ) = 0,7 P ( ) = 0,4 . : ) , . ) 0,1 P ( ) 0 , 4 ) P ( ) 0 , 7

34

.

20. , , P ( ) = : ) , .1 P ( 15 1 ) P ( ) 3

2 2 P ( ) = . 3 5

)

)

2 5

21. A , P P ( ) () 32x 36x + 9 = 0 , : ) ( ) () . ) ( ) ) 1 82

( ) ( )

3 4

.3 8

( )

.3 2 2 5 2 4

22. , , () = Z .

() =

,

) ( ) () . ) ( ) ) 1 4

3 4

.1 2

( )

.

23. (), ( ) = 9 2 12 + 5 , R. ) R. ) () = 1. 24. , .

1 P ( ) P ( ) P ( ). 225. , ,

4 9 2 + = 25 , P ( ) = . P ( ) P ( ) 5

35

.

26. ,

.

(2x 1)( 3x 1)(4x 1) = 0 P(A) , P( ) P( ), P().27. , . ,

]

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

) = 1. ) : i) P ( ) , P( ) , P( ).

ii) P ( )

iii) P ( )

28. 1 2 1 . . , 1 1 . 2 , 2

29. = {1, 2, ... , } . P( ) = , = 1, 2, ... , . . 10 30. A , , : ) P(A B) + 1 = P(A) + P(B) + P( A B ) . ) P(A B) P(A) + P(B) 1. ) P(A) = 0,9 P(B) = 0,8 , P(A B) 0,7. 31. 45% , 55% 25% . , :

) )

) ) .

32. . 40 % ,

28 % .

16 % , :

) . ) . ) .

36

.

33. 50 % 40 % . .

) :.

i) . ii) . iii) . ) : .

34. = { 2, 1, 0, 1, 2 } . :.3

) : 3 7, 3 5 2 , 9 8 , 2 +1 , 6 , , . 2 . ) : 2 , , 0 , 1 , 2 , 1 , , 1,5 . ) : f (x ) = 4x 3 32 x 2 6 x 1 , , x 0 = 1 .

35. , P() + P() 2P( ). f (x) = (x P(A B)) (x P(A B) ) , x R .3 3

) P( ) P( ). ) f x o =4

P(A) + P(B) . 2

) , , f (P(A)) = f (P(B) ) .

36. . ) ( ) = ( ) (). ) ( ) 2 , x R . () = ( ), 5

f (x) = e x

x

.

37. , , P(A) P() . P(A) P() f (x ) = 4x 3 5x 2 + 2x + 2005 , :

) f . ) P(A), P(), P( ), P( ), P( ).

37

.

38. = {1, 2, ... , 10 }

. f (x) = x 3 2x 2 + 3x + 2 , x R . : : f [0 , 5] 68 3 3

1

.

: f [0 , 5] 4 .

) f [0 , 5]. ) X Y. ) . 39. 8 4 c = {1 , 2 , 3 , ... , 20 }, .

: : .

40. . s = 3,8 :

xi

i

20 21 24 26 30

4 2 5 3 6 20

) x . ) X x s , x + s .

(

)

41. 20 . f ( x ) = 2x 2 2( ) x 1 8

.

42. = { 0 , 1 , 2 , 3 , 4 , 5 } . . f (x ) = x 3 2x 2 + 2 x + 1 + 2 , C f f x o = 1 xx.

38

.

43. = { 0 , 1, 2 , 3 , ... , 9 } . . f ( x )= x 3 2x 2 + 4 x + , f .

44. ) f (x ) = e x x , x R , . ) , . N : i ) f (( )) e 1 i i) , P() + e ( ) P(A) + e ( ). 45. 0 , 3x , 2lnx , lnx , 2x . ) .

) x = 1. P ( ) =1

, = 3 , 2 , P(0).

39

.

1) 2) 5) ( ) = ( ) = 1 2 2 3 , ( ) = , 1 4

1 6 1 3

:

, ( ) = 2 9 , ( ) =

:

( ) =

:

) ( 4 ) = ) ( 1 ) =

1 6 1 6

, ( 3 ) = , ( 2 ) =

1 12 2 9 , ( 3 ) =, ( 3 ) = 5 8

5 184 9

, ( 4 ) =

1 3

6) 7) 8) 10) 11) 21) 22) 23) 28) 36)

:

( 1 ) =

2 9 9 40

, ( 2 ) =

1 3

: ) (0) =

) ( ) =

: : :

k = 3 , ( ) =

3 4) ( ) = 0,11 3 , ) ( ) = 2 3

) ( ) = 0, 2) ( ) = 1 6 3 4 ,

) ( ) =

:

) ( ) =

,

) ( ) = 1 2

3 8 3 4

:

) =1 ,2 3 )

( ) =

, () =

:

) =3 4

: :

)

,

1 5

,

)

3 10

,

)

4 5

( ) = 0, 6 ,

( ) = 0,3 40