Кудрявцев А.А. Пособие по теории вероятностей
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P(A) P(B)
A B C
AB BC AC
A B C
1/3
1/6
A B A C
A B ∪ C
A
1
A
2
A
3
P(A
1
A
2
A
3
) = P(A
1
)P(A
2
)P(A
3
)
A
1
, . . . , A
n
P (∪
n
i=1
A
i
) = 1 −
Q
n
i=1
P(
A
i
).
P(A|B) + P(A|B) = 1
P(A|B) > P(B|A) P(A) > 0 P(B) > 0
P(A) > P(B)
n
i p
i
i = 1, . . . , n
n
36
A
B
1
, . . . , B
n
A ⊂ ∪
n
i=1
B
i
P(A) =
n
X
i=1
P(B
i
)P(A|B
i
). (4.1)
B
1
, . . . , B
n
∪
n
i=1
B
i
= Ω
B
i
n ∞
B
i
Ω
A
P(A) =
n
X
i=1
P(AB
i
).
P(A) > 0
P(B
k
|A) =
P(B
k
)P(A|B
k
)
P
n
i=1
P(B
i
)P(A|B
i
)
.
B
k
B
k
8 2
5%
B
k
= { k
} k = 0, 1, 2 B
0
∪B
1
∪B
2
= Ω B
k
B
l
= ∅ k 6= l
A = {
}
P(A)
P(A) = P(B
0
)P(A|B
0
) + P(B
1
)P(A|B
1
) + P(B
2
)P(A|B
2
).
P(A|B
k
) =
10 − k
10
, k = 0, 1, 2,
B
k
10−k
0.05
P(B
0
) = 0.95
2
, P(B
1
) = 0.05 · 0.95 + 0.95 · 0.05, P(B
2
) = 0.05
2
.
P(A) = 0.99
4 2
2
B
k
= {
k } k = 0, 1, 2 A = {
} P(B
2
|A)
P(B
2
|A) =
P(B
2
)P(A|B
2
)
P(B
0
)P(A|B
0
) + P(B
1
)P(A|B
1
) + P(B
2
)P(A|B
2
)
.
P(B
0
) =
C
2
2
C
2
6
=
1
15
, P(B
1
) =
C
1
4
C
1
2
C
2
6
=
8
15
, P(B
2
) =
C
2
4
C
2
6
=
6
15
.
P(A|B
0
) = 0, P(A|B
1
) = 0.5, P(A|B
2
) = 1.
P(B
2
|A) = 0.6
2/5 1/3
5 : 3
A = { } B = {
} C
1
= { }
C
2
= { }
P(C
1
) : P(C
2
) = 5 : 3 P(C
1
) = 5/8 P(C
2
) = 3/8
P(A|C
1
) =
3
5
, P(A|C
2
) =
1
3
,
P(B|C
1
) =
2
5
, P(B|C
2
) =
2
3
,
P(C
1
|A) =
P(C
1
)P(A|C
1
)
P(C
1
)P(A|C
1
) + P(C
2
)P(A|C
2
)
=
3
4
,
P(C
2
|B) =
P(C
2
)P(B|C
2
)
P(C
1
)P(B|C
1
) + P(C
2
)P(B|C
2
)
=
1
2
.
N
1
M
1
N
2
M
2
N
3
M
3
N
1
M
1
N
2
M
2
5 8 10
N
1
M
1
N
2
M
2
n
1
n
2
N M
k λ
k
exp{−λ}/k! λ > 0
p s
N
1
M
1
N
2
M
2
N
3
M
3
7 3
3
5
10 15
1 9
1 5
1 4
1 7
k
1 2
2 3
Ω
1
A
1
A
1
Ω
1
P
1
A
1
A
1
p P
1
(A
1
) = p
n
Ω
i
Ω
1
A
i
A
i
P
1
(A
i
) p
i = 1, . . . , n
P
1
A
δ
1
1
· . . . · A
δ
n
n
=
n
Y
i=1
P
1
A
δ
i
i
(5.1)
(δ
1
, . . . , δ
n
) δ
i
∈ {0, 1} i = 1, . . . , n A
δ
i
i
i δ
i
= 1 δ
i
= 0
n
A
δ
1
1
, . . . , A
δ
n
n
Ω n
ω ∈ Ω
A
δ
1
1
, . . . , A
δ
n
n
(δ
1
, . . . , δ
n
)
{ω}
P
P({ω}) = p
k
(1 − p)
n−k
, (5.2)
k n
(δ
1
, . . . , δ
n
) ξ
n
n P (n, k)
k n
P (n, k) = P(ξ
n
= k) =
X
ω: δ
1
+...+δ
n
=k
P({ω}) = C
k
n
p
k
(1 − p)
n−k
, (5.3)
C
k
n
n k
P (n, k)
ξ
n
n = 1
R(n, k) k
n k = 0, . . . , n
R(n, k) =
n
X
i=k
P (n, i) = 1 −
k−1
X
i=0
P (n, i). (5.4)
R(n, 1)
n
R(n, 1) = 1 − (1 − p)
n
.
1/2
P (4, 3) P (8, 5)
R(4, 3) R(8, 5)
P (4, 3) = C
3
4
1
2
4
=
1
4
>
7
32
= C
5
8
1
2
8
= P (8, 5),
R(4, 3) =
4
X
i=3
P (4, i) =
5
16
<
93
256
=
8
X
i=5
P (8, i) = R(8, 5).
n = 200
p = 0.01
P (200, 3)
P (200, 3) = C
3
200
(0.01)
3
(0.99)
197
.
P (200, 3) ≈ 0.18136
n = 22 500
p = 0.2
ξ
n
4380 4560
P
P =
4560
X
k=4380
P (22500, k) =
4560
X
k=4380
C
k
22500
(0.2)
k
(0.8)
22500−k
.
P (n, k)
P (n, k)
P (n, k) =
(
C
k
n
p
k
(1 − p)
n−k
k = 0, . . . , n;
0 k = n + 1, n + 2, . . . .
n n p
n