Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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A P (A)
A
H
(Ω, F, P) Ω
H F
H
P
(A)
A ∈ F
Ω
H
Ω
F , 2
Ω
ω ∈ Ω p(ω) > 0
P
ω∈Ω
p(ω) = 1
p(ω) ω
A ∈ F
P
(A) ,
P
ω∈A
p(ω).
0 6 P (A) 6 1 P (Ω) = 1
A ∩ B = ∅ P (A ∪ B) = P (A) + P (B)
P (A
c
) = 1 − P (A) P (∅) = 0
A ⊂ B P (A) 6 P (B) P (B \A) = P (B) −P (A)
A =
∞
S
n=1
A
n
A
i
∩ A
j
= ∅ i 6= j
P (A) =
∞
P
n=1
P (A
n
).
H
A
A
H
P
(A) =
|A|
|Ω|
,
|A| |Ω|
A Ω
2
Ω = {(a
1
, a
2
)| a
i
= 1, 6, i = 1, 2}.
Ω 6
2
A
8
{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}
P
(A) =
|A|
|Ω|
=
5
6
2
n
4
n
A = { } B = {
}
n
n 1, 2, . . . , n
k k
a
i
i
ω
k
ω
(a
1
, . . . , a
k
) 1 6 a
i
6 n i = 1, . . . , k
|Ω| = n
k
k
(b
1
, . . . , b
k+n−1
) k n −1
(b
1
, . . . , b
k+n−1
)
(n − 1)
n
Ω = {ω = (b
1
, . . . , b
k+n−1
)| b
j
= 0 1,
X
j
b
j
= n − 1}.
k
n
Ω n − 1
k + n −1 n −1
|Ω| = C
n−1
k+n−1
n 1, 2, . . . , n
k
k
a
i
i
ω
k
ω
(a
1
, . . . , a
k
) 1 6 a
i
6 n i = 1, . . . , k a
i
6= a
j
i 6= j
k
n
|Ω| = A
k
n
k
{a
1
, . . . , a
k
} 1 6 a
i
6 n i = 1, . . . , k
a
i
6= a
j
i 6= j
k
n |Ω| = C
k
n
4 3 5
12
3 C
k
n
n = 12, k = 3 |Ω| = C
3
12
m
1
= (3 + 5)C
2
4
m
2
= (4 + 3)C
2
5
m
3
= (5+4)C
2
3
|A| = m
1
+m
2
+m
3
P (A) =
|A|
|Ω|
=
8C
2
4
+ 7C
2
5
+ 9C
2
3
C
3
12
.
α
a b
k
a b a > 2 b > 2
2n 2n
2n
5
25
5
1, 2, 3, 4, 5
1, 2, 3, 4, 5
n n
a b
c d
K N M
k n
m
|Ω| = 36!
33!
4!
|A| = 4!33!
P (A) =
4!33!
36!
n
m
n+k
n
m 6 n
n
n n + k |Ω| = C
n
n+k
m
m
n + k − m
n − m C
n−m
n+k−m
|A| = C
n−m
n+k−m
P (A) =
C
n−m
n+k−m
C
n
n+k
.
2n
n
m k
r
k
n |Ω| = C
k
n
A
i
i k
(k − i)
i
C
i
m
k −i
C
k−i
n−m
|A
i
| = C
i
m
C
k−i
n−m
P (A
i
) =
C
i
m
C
k−i
n−m
C
k
n
A
i
A = A
r
∪ A
r+1
∪ ··· ∪ A
k
.
P (A) =
k
X
i=r
C
i
m
C
k−i
n−m
/C
k
n
.
n m
k
r
k
1
10000
F Ω
σ F
Ω ∈ F
A ∈ F A
c
= Ω \ A ∈ F
(A
n
)
n∈N
∈ F
∞
S
n=1
A
n
∈ F
F σ Ω (Ω, F)
(Ω, F,
P
) Ω
F σ Ω
P
: F 7→
[0, 1]
P
(Ω) = 1
(A
n
)
n∈N
⊂ F A
i
∩ A
j
= ∅ i 6= j
P
(
∞
[
n=1
A
n
) =
∞
X
n=1
P
(A
n
).
Ω
H
A ∈ F
H P (·)
P (A)
A ∈ F
(Ω, F, P )
P
(∅) = 0
A, B ∈ F A ∩ B = ∅
P
(A ∪ B) =
P
(A) +
P
(B)
A, B ∈ F
P
(A ∪ B) =
P
(A) +
P
(B) −
P
(A ∩ B)
A, B ∈ F A ⊂ B
P
(A) 6
P
(B)
P
(B \ A) =
P
(B) −
P
(A)
P
(A
c
) = 1 −
P
(A)
(A
n
)
n∈N
⊂ F
P
(
∞
S
n=1
A
n
) 6
∞
P
n=1
P
(A
n
)
A
n
∈ F A
n
⊂ A
n+1
n ∈ N lim
n→∞
P
(A
n
) =
P
(
∞
S
n=1
A
n
)
A
n
∈ F A
n
⊃ A
n+1
n ∈ N lim
n→∞
P
(A
n
) =
P
(
∞
T
n=1
A
n
)
R
1
< R
2
< . . . <
R
10
A
i
R
i
A
1
∪ A
3
∪ A
6
A
2
∩ A
4
∩ A
6
∩ A
8
(A
1
∪ A
3
) ∩ A
6