элементы теории вероятностей

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  • : n!

  • , .: 17

  • n - 1 n n! = 12 3 ... (n 2)(n 1)n0! = 11! = 12! = 12 = 23! = 123 = 6

    4! = 1234 = 245! = 1234 5 = 1206! = 1234 5 6 = 720:5! : 3! = 4! : 6! = 15! 16 = (n - 1)! n =:4 5 = 201 : (5 6) =1/ 3016!n!

  • n , . . 0!=1

    n = n!3 = 3! =123 = 6

  • : 4 ? 7 . ? , 5, 7, 8, , . , ? , 1, 3, 5, 7 ( )4! = 247! = 50403! = 61+3+5+7 = 16 4! = 24 16 24 = 384

  • 5) ( ), 3, 5, 7, 9, , : ) 3? ) 15?6) , , , ?7) 12 , 5 , , ?8) 5 5 1 10, , ?) 13! = 1123 = 6 ( 3 , 123)) . . 15, 3 5. 3+5+7+9=24 3, 5, 5, . . 5 1 , 3! : 6 , , 3!=6 -- 3!=6 6 6 = 36 5!=1202) 7 8!= 403203) 120 40320 = 4 838 4005! 5! = 120 120 = 14 400

  • m n, (n < m), ,

    24 = 12

    4 2 12

    nm = m(m-1)(m-2)... n

    24 = 43 = 12(4 3)

    nm = m!/(m n)!

    24 = 4!/(4 2)! = 24/2 = 12mm = m!

  • : - , 30 . ? 1, 2 3 8 100 ? 6 . 2 ; 4 ; 6 ? ( ) 0, 1, 2, 3, 4, 5, 6? , ?

    34 = 4 3 2 = 24 230 = 30 29 = 870 38 = 8 7 6 = 336 26 = 6 5 = 30 46 = 6 5 4 3 =36066 = 6! = 720

    1. 37 = 765 = 210 ;2. 26 = 65 = 30 , ;3. 210 30 = 180 710 69 = 10987654 - 987654 =987654(10 1) = 544320

  • m n, (n < m)

    nm m n

    m n

    .

    :, ,

    24 = 6

  • nm = nm / n

    C24 = 24 / 2 =(43)/2 = 12:2 = 6

    nm = m!/(n!(m n)!)

    C24 = 4!/(2!(4 2)!) =34/2 = 12:2 = 6

    :nm = mm n , n> m

    Cmm = m! / (m!(m m)!)=10! = 1

  • : , 30 ? , 78 ? 5 36?4. , 80 3 ?5. 16 12 . 4 3 . ?C330 = 30!:(3!(30 3)!) =(282930):6 = 4060C278 = 78!:(2!76!) =(7778):2 = 3003C536 = 36!:(5!31!) =(3233343536):(2345) = 376992

    416 312 = 400 400

  • , ,

  • : , , : ) ; ) ; ) ; ) ; ) ; ) ; ) .,,, , 2. ? ) - ; ; ) - ; ; ) 5; 7; ) ; 73., ,

  • 3. : ) ; ) , ; ) 5 .4. : ) , 2 3 ; ) ; ) .5. : ) ; ) ; ) - ; ) 12 12 . I, II, III, IV7 6 ,

  • , ()probabilite

    ()()~m/nm n

    ()() = m/nm n

    .

    4040 0,5070

    4092 0,5005

    10 000 0,4979

    20 480 0,5068

    24 000 0,5005

  • : , 3, ? (1717-1783): , .

    3. 25 11 8 . , , ) ; ) ?

    4. 36 3 . , ) ; ) ?

    n = 6, m = 2, () = 2 : 6 = 1/3 n = 4, m = 11 - 2 () = 1 : 4 = 0,25 = 25%

    ) n = 363, m = 353 , () = 11/12) () + () = 1,

    () = 1 11/12 = 1/12) n = 25, m = 25 (8 + 11) = 6, () = 6 : 25 = 0,24 = 24%) n = 25, m = 8 + 11 = 19, () = 19 : 25 = 0,76

  • , , (/) , , (/) = () : ()

    ()

  • : 30 . : , . : , 10. ? :n = 30 ,m = 10 , 3,

    () = m/n = 10/30 = 1/3 , . . , , 3, 10.n = 20 , 10,m = 10 3 = 7 , 10, 3.() = m/n = 7/20, (/) = 7/20(/) > () () = 7/30() = 20/30(/) = 7/30 : 20/30 = 7/20

  • , +

    1+2+...+n n

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

  • , , .

    () = ()()(N) = ()()()(N)

  • 1, 2, 3, 1, 2, 3, () = (/1)(1) + (/2)(2) +

    2003 04,2004 - 05,2005 - 06