Metody prob 1


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(W, , ) W
s W
s W
W
ó ó
" W -

, , ,...
1 2 3 U
s
, , ,...
1 2 3 I
, -
Ć
ć
ó
= W -
I U

Ł ł
, ,Ć,Ć,Ć,.....
ó
Ć = W
W s W
:
+
(W) = 1
1
, = 1,2,3,... = Ć ą
Ą Ą
ć
= ( )
U

=1
Ł =1 ł
W W = {w1,v ,...,v } = 2W
2
1
( ) = ({w }) =
W
W = {1,2,3,4,5,6}
{2,4,6} 3
( ) = = = 0,5
6
{1,2,3,4,5,6}
(W, , ) , , , ,...,
1 2
(Ć) = 0
( )[0,1]
ć
, = 1,2,3,..., = Ć ą = ( )
U

=1
Ł =1 ł
,
( ) Ł ( )
, = 1,2,3,...
Ą Ą
ć
Ł ( )
U

Ł =1 ł =1
, = 1,2,3,...,
ć
Ł ( )
U

=1
Ł =1 ł
2
Ą
ć
= lim ( )
+1 U

Ą
Ł =1 ł
Ą
ć
= lim ( )
+1 I

Ą
Ł =1 ł
, ( - ) = ( )- ( )
ó
( ) = 1- ( )
, ( ) = ( )+ ( )- ( )
, ,
( ) = ( )+ ( )+ ( )- ( )- ( )- ( )+ ( )
(Ć) = (Ć Ć Ć .....) = (Ć)+ (Ć)+ (Ć)+ ........ (Ć) = 0
(W) = 1
, = 1,2,3,..., = Ć ą >
= Ć
Ą Ą
ć ć
= = ( ) = ( )
U U

=1
Ł =1 ł Ł =1 ł =1
,
( ) = ( ( - ))
( ) = ( )+ ( - ) ( - ) ł 0
, = 1,2,3,...
= , = - ( ... )
1 1 1 -1
Ą Ą
=
U U
=1 =1
( ) Ł ( )
Ą Ą Ą Ą
ć ć
= = ( ) Ł ( )
U U

Ł =1 ł Ł =1 ł =1 =1

+1
= , = - ,...., = - ,....
1 1 2 2 1 -1
=
U
=1
Ą Ą Ą
ć ć ć
= = ( ) = lim ( ) = lim = lim ( )
U U U

Ą Ą Ą
Ł =1 ł Ł =1 ł =1 =1 Ł =1 ł
,
( ) = (( - ) ) = ( - )+ ( ) - ,
3
,
( ) = ( ( - )) = ( )+ ( - ) = ( )+ ( - ( )) = ( )+ ( )- ( )
W l
2 3
W
l ( )
( ) =
l (W)
(0,2)(0,2) W l2(W) = 4
1 1
= {( , ) W : + < 1} l2( ) = ( ) =
2 8
= Ć
( ) = ( ) ( )
4
2 1
W = {( , ),( , ),( , ),( , )} ( ) = = =
4 2
W
2 1 1
( ) = = = ( ) = = ( ) = ( ) ( )
4 2 4
W W
( ) = ( ) ( ) ( )
( ) > 0
( )
( / ) =
( )
W = {( , ),( , ),( , ),( , )}
1
1
4
({( , )}/{( , ),( , )}) = =
1
2
2
1
1
4
({( , )}/{( , ),( , ),( , )}) = =
3
3
4
5
W
, , ,....
1 2 3
W
= W, = Ć , ą
U
{ , ,...., } W
1 2
( ) = ( / ) ( )

=1
ć

( ) = ) = ( ) = ( / ) ( )
U(

=1 =1
Ł =1 ł
1
6
2
3
3
1 1 1 1 1
( ) = ( / ) ( ) = 0,03 + 0,05 + 0,02 = 0,1 =

3 3 3 3 30
=1
{ , ,...., } W
1 2
( / ) ( )
( / )=
) ( )
( /
=1
( ) ( / ) ( )
( / )= =
( )
) ( )
( /
=1
1
2
3
7
1
0,05
( / ) ( )
2 2 3
( / ) = = = 0,5
2
3
1
) ( )
( /
30
=1
ć
-
( ) =

Ł ł
( )
= 1-
8
= { , , , , , , , , , }
1 2 1 2 3 1 2
W = {( , ,..., ), ą , }
1 2 10
W
W = 10!
= 2!3!2!= 24
24 1
( ) = =
10! 151200
= { ,..., , ,..., }
1 10 1 15
W = {{ , , }, ą , }
1 2 3
W
25
ć
W =

3
Ł ł
9
10 15
ć ć
=

2 1
Ł łŁ ł
10 15
ć ć


2 1
Ł łŁ ł
( ) =
25
ć


3
Ł ł
W = {( , , ), ą , }
1 2 3
W
!
W =
( - 3)!
10 15
ć ć
= 3!.

2 1
Ł łŁ ł
10 15 10 15
ć ć ć ć
3!

2 1 2 1
Ł łŁ ł Ł łŁ ł
( ) = =
!
25
ć


( - 3)!
3
Ł ł
W = {( , ), { , , }}
1 2
W = 32 = 9
1
= {( , ),( , ),( , )} = 3 ( ) =
3
10
W = {( , , , ), , , , , + + + = 15}
1 2 3 4 1 2 3 4 1 2 3 4
15 + 4 -1
ć
W =

4 -1
Ł ł
= {(1+ ,1+ ,1+ ,1+ ), , , , , + + + = 11}
1 2 3 4 1 2 3 4 1 2 3 4
11+ 4 -1
ć
=

4 -1
Ł ł
11+ 4 -1
ć


4 -1
Ł ł
( ) =
15 + 4 -1
ć


4 -1
Ł ł
= {3,4,5,7,9}
( ) = 0,5; ( ) = 0,2; ( ) = 0,4; ( ) = 0,2; ( ) = 0,1; ( ) = 0,1; = Ć
11

( ) = ( )+ ( )+ ( )- ( )- ( )- ( )+ ( )
( ) = 0,5 + 0,2 + 0,4 - 0,2 - 0,1- 0,1+ 0 = 0,7
ć ć ć
( )ó ( )ó ( )ó

Ł ł Ł ł Ł ł
ć =
( )ó ć ( )ó ć ( )ó
ć
Ł ł Ł ł Ł ł
Ł ł
ć + ć + ć =
( )ó ć ( )ó ć ( )ó
ć
Ł ł Ł ł Ł ł
Ł ł Ł ł Ł ł
( )- ( )- ( )+ ( )- ( )- ( )+ ( )- ( )- ( ) =
0,5 - 0,2 - 0,1 + 0,2 - 0,1- 0,1+ 0,4 - 0,2 - 0,1 = 0,3
ć
ó ó ó
( ) = ( )ó = 1- ( ) = 0,3

Ł ł
( ) ł ( )+ ( )-1
1 1
( ) = ( ) = , ( \ ) = ( \ ) ( ), ( \ )
2 4
1 3
( ) = , ( ) = , = Ć
4 4
ó ó
( ), ( ), ( )
( ) = 0,4 ( ) = 0,6
( / )
( )
ó
( )
( ) = ( ) ( ) = 0,24
( ) = ( ) ( )
2
( ) = = = ( )
= 0 = 1
12
W = 3!= 6
ó
1
2
2
ó
( ) = ( ) =
1 2 3
( )+ ( )+ ( )- ( )- ( )- ( )- ( ) =
1 2 3 1 2 2 3 1 3 1 2 3
2! 2! 2! 1 1 1 1 1 1 4
+ + - - - + = 3 - 2 =
3! 3! 3! 3! 3! 3! 3! 3 6 6
2 1
ó
( ) = 1- ( ) = =
6 3
[- 2,3]
[- 2,3] W = [- 2,3]
l1( ) l1( ) l1( )
( ) = = =
l1(W) 3 - (- 2) 5
l1( ) 3
= { [- 2,3]: > 0}= (0,3] ( ) = =
5 5
l1( ) 2
2
= { [- 2,3]: < 1}= (-1,1) ( ) = =
5 5
13
[-1,2]
[0,5]
W
2
l2(W) = P(60) = 3600P
l2( ) 3481P 3481
2
l2( ) = P(59) = 3481P = =
l2(W) 3600P 3600
( Ł )
1 2
W = {( , , ), { , }} ({ }) = , ({ }) =
1 2 3
3 3
14
1 1 1 1 2 1 1 2 1 2 1 2
({( , , )}) = = ({( , , )}) = = ({( , , )}) = =
3 3 3 27 3 3 3 27 3 3 3 27
1 1 2 2 2 2 1 4 1 2 2 4
({( , , )}) = = ({( , , )}) = = ({( , , )}) = =
3 3 3 27 3 3 3 27 3 3 3 27
2 1 2 4 2 2 2 8
({( , , )}) = = ({( , , )}) = =
3 3 3 27 3 3 3 27
6
( ) =
27
5 1
ć
W = {6, 6, 6, 6, 6, 6,...}, {1,2,3,4,5} ({ ... 6}) =

6 6
Ł ł
5 1 5
2 -1
Ą
5
6
= { 6, 6, 6,...} ( ) =

ć 5 1 = 6ć 5 = 36 = 11
2
11
6 6
Ł ł
=1
1-

36
6
Ł ł
1 5 25 125
= {6, 6, 6, 6} ( ) = + + +
6 62 63 64
15
W = {( , ), , {1,2,3,4,5,6}} W = 36
6
= {(6, ), {1,2,3,4,5,6}} = 6 ( ) =
36
12
= {( ,1),( ,2), {1,2,3,4,5,6}} = 12 ( ) =
36
6
= {(6,1),(5,2),(4,3),(3,4),(2,5),(1,6)} = 6 ( ) =
36
2
= {(6,1),(6,2)} = 2 ( ) =
36
1
= {(6,1)} = 1 ( ) =
36
2
= {(6,1),(5,2)} = 2 ( ) =
36
1
= {(6,1)} = 1 ( ) =
36
( ) = ( ) ( ) ( ) = ( ) ( ) ( ) = ( ) ( )
( ) ą ( ) ( ) ( )
W = {( , ), { , }}
1 2
1
( ) 1
4
( / ) = = =
3
( ) 3
4
16
1
=
5
W1 = {{ , }, , { , , , , 1, 2 ,..., 6}}
1 2 3 4
0
6
ć

2 5

Ł ł
( ) = =
0
10 15
ć


2
Ł ł
1
4 6
ć ć

11 8

Ł łŁ ł
( ) = =
1
10 15
ć


2
Ł ł
2
4
ć

2 2

Ł ł
( ) = =
2
10 15
ć


2
Ł ł
17
2
= Ć, ą , = W
U
=0
/
0
/
1
/
1
8
( / ) =
0
10
9
( / ) =
1
10
( / ) = 1
2
2
8 5 9 8 2 22
( ) = ( / ) ( ) = + +1 =

10 15 10 15 15 25
=0
2
( / ) ( ) 5
2 2 15
( / ) = = =
2
22
( ) 33
25
18
19


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