Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
.pdf
P (A1A2 . . . An) = P (A1)P (A2) . . . P (An).
A1, A2, . . . , An |
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A1 |
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, . . . , An |
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P (A) = 1 − P (A1)P (A2) . . . P (An). |
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A |
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A1 A2 . . . An |
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A = A1A2 |
. . . An. |
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A1, A2, . . . , An |
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A = A1A2 |
. . . An |
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P (A) = P (A1)P (A2) . . . P (An) |
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P (A) = 1 − P (A) = 1 − P (A1)P (A2) . . . P (An). |
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A1, A2, . . . , An 
p
A
P (A) = 1 − (1 − p)n.
A 
n 
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P (A) = 1 − (1 − 0, 4)n. |
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1 − 0, 6n 0, 9 |
P (A) 0, 9 |
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n lg 0, 6 lg 0, 1 |
lg 0, 6 < 0 |
n 5 |
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n lg 0, 1/ lg 0, 6 |
lg 0, 1/ lg 0, 6 ≈ 4, 5 |
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H
A 
A 
H
H
H
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P (A) |
P (Hi/A) = |
P (Hi)P (A/Hi) |
= |
P (Hi)P (A/Hi) |
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n |
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k |
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P (A) |
P (Hk)P (A/Hk) |
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=1 |
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P (H2/A) = |
P (H2) · P (A/H2) |
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0, 6 · 0, 8 |
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P (H1) · P (A/H1) + P (H2) · P (A/H2) |
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· 0, 6 + 0, 6 · 0, 8 |
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0, 4 |
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P (H2) = 3/5 |
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H2 |
H1 |
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A 
B 
P (A + B) = P (A) + P (B) = 306 + 308 = 157 .
P (A + B) = 7/15
A 
B 
P = P (A+ B) 
A 
B 
P = P (A) + P (B) 
C4020
C3720
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P (A) = C3720/C4020. |
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P (B) = C31 · C3719/C4020. |
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P = |
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(C3720 + C31 · C3719) = 0, 5. |
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C20 |
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40 |
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P (A) = 0, 5
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n |
−(5/6)n |
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1 − (5/6) |
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> 0, 75 |
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1 −n(5/6)n |
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(5/6) < 0, 25 |
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n ln |
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< ln |
1 |
, n > |
ln 4 |
≈ 7, 6. |
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4 |
ln 1, 2 |
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n 8 
n 8
P (A1A2A3) = P (A1) · P (A2) · P (A3)
P1 = (11/25)3 ≈ 0, 085.
P (A1A2A3) = P (A1) · P (A2/A1) · P (A3/A1A2)
P2 = |
11 |
· |
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≈ 0, 072. |
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24 |
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P1 ≈ 0, 09 
P2 ≈ 0, 07
p1 = 0, 05; p2 = 0, 03; p3 = 0, 02,
qi = 1 − pi
q1 = 0, 95; q2 = 0, 97; q3 = 0, 98.
P1 = q1q2q3 = 0, 95 · 0, 97 · 0, 98 = 0, 903
P2 = 1 − q1q2q3 = 0, 097
P (A + B + C) = P (A) + P (B) + P (C) −P (AB) −P (AC) −P (BC) + P (ABC)
A, B, C
P = p1 + p2 + p3 − p1p2 − p1p3 − p2p3 + p1p2p3 = 0, 097.
P1 ≈ 0, 90 
P2 ≈ 0, 10
P (A) = 1 · 3951 · 2650 · 1349 ≈ 0, 106;
P (B) = 1 · 3952 · 2652 · 1352 ≈ 0, 094.
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C |
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25 |
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24 |
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≈ 0, 945. |
P (C) = 1 − P (C) = 1 |
− |
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52 |
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P (A) ≈ 0, 11 
P (B) ≈ 0, 09 
P (C) ≈ 0, 95
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p1 = 0, 8 p2 = 0, 85 |
p3 = 0, 9 |
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q1 = 0, 2; q2 = 0, 15; q3 = 0, 1 |
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P1 = p1q2q3 + q1p2q3 + q1q2p3 = 0, 056. |
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P2 = p1p2q3 + p1q2p3 + q1p2p3 = 0, 363; |
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P3 = p1p2p3 |
= 0, 612; |
P4 |
= q1p2q3 = 0, 017; |
P5 = p1p2q3 |
= 0, 068; |
P6 |
= q1q2q3 = 0, 003; |
P7 = 1 − q1q2q3 = 0, 997. |
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P1 ≈ 0, 06 
P2 ≈ 0, 36 
P3 ≈ 0, 61 
P4 ≈ 0, 02 
P5 ≈ 0, 07 
P6 ≈ 0, 00 
P7 ≈ 1, 00
p = 0, 0003.
p1 = 0, 02 |
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P = 1 −nqn |
q = 0, 9997, n = 300 |
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q |
P ≈ 0, 067. |
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1 − 0, 9997n 0, 02, |
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0, 9997n 0, 98, n · lg0, 9997 lg0, 98, n |
lg0, 98 |
, n 88. |
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lg0, 9997 |
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n 
n lg(1 − p1)/lg(1 − p).
P ≈ 0, 07 
n 88
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P (A) = 1 − P (A) |
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C |
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65 |
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P (A) = 1 − |
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= 1 − |
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C9 |
76 |
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¯
A
P (A) = 65/76
p1 = 0, 03 
m 
n 
n 
m 
k 
A
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n |
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A |
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− p = q |
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Pn(m) |
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P (A) = p P (A) = 1 |
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A |
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n |
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A |
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A |
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A |
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AAAA |
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A |
m |
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n−m |
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n |
m |
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Cnm = |
n! |
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m!(n − m)! |
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A |
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A |
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A |
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B1 = AA . . . AAA . . . A |
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m |
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A/n − m |
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P (B1) = pmqn−m,
P (B2) = · · · = P (Bk) = pmqn−m,
k = Cnm
P |
(m) = P (B |
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+ |
· · · |
+ B |
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· · · |
+ P (B |
) = kpmqn−m = Cmpmqn−m. |
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Pn(m) = Cnmpmqn−m, |
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Cnm = |
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n! |
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m!(n − m)! |
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n = 8 m = 5 p = 0, 6 q = 1 − 0, 6 = 0, 4 |
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P8 |
(5) = |
8! |
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0.65 · 0.43 ≈ 0.28. |
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5!(8 |
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5)! |
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P8(3 m 5)
P8(3 m 5) = P8(3) + P8(4) + P8(5).
P8(3 m 5) = C83 · 0, 63 · 0, 45 + C85 · 0, 65 · 0, 43 ≈ 0, 12 + 0, 23 + 0, 28 = 0, 63.
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n |
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P (A) = Cnp |
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P (A) = 1 − P (A) = 1 |
− (1 − p) |
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− p) |
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= 0, 4 |
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≈ 0, 001 = P (A) ≈ 0, 999. |
P (A) = (1 |
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A A
n 
A1, A2, . . . , Am |
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p1, p2, . . . , pm |
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m |
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pi |
= 1. |
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=1 |
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m |
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k1 |
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A1 |
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km |
Am j=1 kj = n |
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Pn(k1, k2, . . . , km) = |
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n! |
· p1k1 |
· p2k2 . . . pmkm . |
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k1!k2! . . . km! |
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