.
- , , - .
, - , , , ½, , . , , - ( ) . " " . ( ) , , , . , , , .
.
S , , :
) S . , , , , , , .
) S P (A / S), . , , , , - N .
n ( n S) h = m/n m , , n. S , , , .
, , (), , . , (, 0,515). 19 . 1- 20 . , , ..
, , , , (. ). .
.
. , , , E1, E2,..., ES ( , ). . Ek - . pk . , , " Ei, Ej,..., Ek". Ei, Ej,..., Ek , () , :
P (A) = pi + ps + + pk. (1)
p1 = p2 =... ps = 1/S
() = r/s. (2)
(2) , - r , , s "" . "" "", .
. 36 (i, j), i - , , j - . . - " 4", (1; 3), (2; 2), (3; 1). , (A) = 3/36 = 1/12.
- , : () (). A 1, A 2,..., Ar,-, : " A1, 2,..., Ar".
A1, .2,..., Ar, : " A1, A2,..., Ar". È, - Ç. , :
B = A1 È A2 È È Ar, C = A1 Ç A2 Ç Ç Ar.
, , , .
. . - .
. A1, A2,..., Ar , , .
, - " 4", A2, A3, A4, , 2, 3, 4. 1/36; 2/36; 3/36. ()
1/36 + 2/36 + 3/36 = 6/36 = 1/6.
, , . A1, A2,..., Ar , , - , ""
. A1, A2,..., Ar A1, A2, , 1 ,..., Ar , A1, A2,..., Ar-1 . :
P (A1 Ç A2 Ç Ç Ar) = P (A1) P (A2) P (Ar), (3)
. (3) , .
. 4 0,2 . . ?
[., (, , , ) , (), ()]. 2222 = 16 . (3) . , (, . , ) 0,20,80,80,8 = 0,1024; 0,8 = 1-0,2 - . " " (, , , ), (, , , ), (, , , ). (, , , ), :
0,20,20,20,8 =...... =0,80,20,20,2 = 0,0064;
,
40,0064 = 0,0256.
, : A1, A2,..., An , m
Pn (m) = Cnmpm (1 - p) n-m; (4)
Cnm n m. n (4) . 100, , 8 32. (4) ,
0,0009. , 8 £ m £ 32 . , .
: A1, A2,..., Ar ,
. , T1, T2,..., Tn-1, Tn, Ai, Bj,..., Xk, Yl T1, T2,..., Tn-1, Tn.
P (Ai), P (Bj/Ai), , P (Yl/Ai Ç Bj Ç Ç Xk). (5)
(5) () , , ( , ). : ) , (5) P (Ai), P (Bj),..., P (Yl); ) - , (5) : P (Ai), P (Bj /Ai),..., P (Yi / Xk). , . , , (i) P (Bj / Ai),..., P (Yl / Xk)
. Er ,, , X. x1, 2,......, xs ; r = 1, 2,..., s . . , (i, j) = i + j - . 2, 3, 4,..., 11, 12; 1/36, 2/36, 3/36,..., 2/36, 1/36.
,
{X1 = x1}, {X2 = x2}, , {Xn = xn}, (6)
xk - - Xk. , xk (6) . , , a < X1 + 2 +... + Xn < b ..
. .
, , .. .
. , , , - , .. . , - (, ), .. , , .
, (2). , , (, , ), . , , , .
1933 . . . . - - U u, . . (A),
1. 0 £ () £ 1,
2. P (U) = 1,
3. A1,..., An - ,
() = (A1) + P (A2) + + (An).
, 3 . . , , . . , - .. . , , , . , , ..
.
, , . . . . , , - , , , . , ,
X1, 2,..., Xn,... (7)
- , EXk = , DXk = s2 Yn - n (7):
Yn = (X1 + X2 + +Xn)/n.
, e > 0, |Yn - a| £ e n ¥ 1, , , Yn , . , , Yn 0 s2 / n. , Yn n Xn, .
20- . 20 . , , . , , X1 , 2 - .., X1 +... + Xn ( n-) n 1/a ( - , 1) . , n- , , n 1/a, n ( n).
.
.
, , , , . , . . . (t). t , , , , . , t , . , , X (t1), X (t2),..., X (tn) t1, t2,..., tn n > 0. .
. (t) , t0 t1 (t0 < t1) X (t1) , (t) t £ t0, X (t0) ( ). , . t0 ; t0 t > t0, t0 .
. , , (,
z (l) ). .
. , , . .
.
17 . , . . X. , . . , ( 1713).
() (18 . 19 .) . (), . (), . () . (). - , ( , , ). , (1812) (1837); . (, 1806) (1808) .
. (2- 19 .) . . , . . . . (). ( 18 . . . . , . . ; . . , , . . , ). 2- 19 . . , . (1867) . (1887) . (1901) . (1907) , .
2- 19 . ( - . , - . ) ( - . ), , () . , , , ( ) , . ( - . , . , . , - . , - . , . , . , - . ) , . . . , , . . . . . . ( 30- .) ( . . ) . . . () . . () . , . . - .
. . . ., , 3 ., . - ., 1952;
. ., , 4 ., ., 1965
., ( ), . ., 2 ., . 1-2, ., 1967.
. ., , 4 ., . - ., 1946
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