Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
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n |
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Cnm = |
n! |
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m!(n |
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− m)! |
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Cnm |
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n |
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m (Anm) m! |
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m |
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m |
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m |
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Anm |
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An |
= Cn |
· m! = |
Cn |
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m! |
m!(n |
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Cnm = Cnn−m 
Cn0 = Cnn = 1, Cn1 = Cnn−1 = n
n
Cni = 2n
i=0
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n |
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i |
(a + b)n = |
Cni aibn−i. |
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a = b = 1
n 
n = C105 |
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10! |
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10! |
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= 252. |
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· (10 − 5)! |
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5! |
5! |
· 5! |
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m
U P 
A
0 ( ) 1 ( ¯) = 1 − ( ) p A , p A p A
n = 75
A
m = 64 
P (A) = 64/75.
B 
m = 11 
P (B) = 11/75
n 
A 
m 
=
B 
C53 |
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P (B) = |
C53 |
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3 · 4 · 5 |
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1 |
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C803 |
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78 · 79 · 80 |
5056 |
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P (A) ≈ 0, 82; P (A) = 1/5056 |
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N = 100 |
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M = 12 |
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n = 10 |
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m = 2 |
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C10010 |
A |
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C122 |
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C10−2 |
= C8 |
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100−12 |
88 |
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A |
C122 · C888 |
P (A) = |
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C122 · C888 |
≈ |
0, 245. |
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C10010 |
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P (A) ≈ 0, 25 |
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107
A710 = 10!/3!
P (A) = 10!/(3! · 107) = 189/55 ≈ 0, 061. P (A) ≈ 0, 06
n = 75 |
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m = 7 |
P (A) = 1/74 |
m = 1, P (B) = 1/75 m = C75 = 21 |
P (C) = 21/75 = 3/74 |
C71 = 7 |
C53 |
C61 = 6 |
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m = C53 · C71 · C61 |
P (D) = 60/74. |
P (A) = 1/74 
P (A) = 1/75 
P (A) = 3/74 
P (A) = 60/74
P (A) = 1/A56 = 1/720.
P (A) = 1/720
n = 8!
m = 6 · 3! · 5! P (A) = 6 · 3! · 5!/8! = 283 . P (A) = 383
n 
P (A1 + . . . + An) = P (A1) + . . . + P (An), 
AiAj = V 
i = j.
P (A + B) = P (A) + P (B) − P (AB).
n
m1 
m2 m
m
1
A
m
m2 B
m1 + m2 − m
P (A + B) = m1 + m2 − m = m1 + m2 − m = P (A) + P (B) − P (AB). n n n n
A 
B
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P (A + B) |
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P (A) = |
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, P (B) = |
1 |
, P (A · B) = |
1 |
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4 |
9 |
36 |
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1 |
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P (A + B) = |
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4 |
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36 |
3 |
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P (A/B) = PB (A)
A 
B 
P (A/B) = 3/5.
A 
P (A) = 36 ,
P (AB) = P (A) · P (B/A).
n 
m1 
A
m
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m1 |
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B |
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P (A) = m1/n, |
P (A · B) = m/n, P (B/A) = m/m1. |
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P (AB) = |
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m |
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= |
m1 |
· |
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= P (A) · P (B/A). |
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m1 |
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P (A1A2 . . . An) = P (A1)P (A2/A1)P (A3/A1A2) . . .
. . . P (An/A1A2 . . . An−1).
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B |
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A |
A |
B |
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P (B/A) = P (B). |
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= P (A) · P (B/A) = P (A) · P (B) |
P (A · B) = |
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P (A · B) = P (B · A) = |
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= P (B) · P (A/B) |
P (A) · P (B) = P (B) · P (A/B) P (A/B) = P (A) |
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A |
B |
P (A · B) = P (A) · P (B).