Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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A
1
A
3
A
6
A
1
A
3
A
6
R
i
i = 1, . . . , 9
A
6
A
1
∪ A
3
∪ A
6
= A
6
A
2
A
4
A
6
A
8
R
2
A
2
∩ A
4
∩ A
6
∩ A
8
= A
2
.
(A
1
∪ A
3
) ∩ A
6
= A
3
∩ A
6
= A
3
A
k
k k = 1, 2, 3, 4, 5, 6
A
k
A
B
C
D
A B C D
A
B
C
D
A ∪ B B ∩ C C∆D (B ∪ D) \ C
A
B
A4B
c
A
c
4B A
c
∩ B
c
B
c
\ A A
c
\ B
n A
i
i
1 6 i 6 n A
i
k 1 6 k 6 n
k k
n
1, 2, . . . , n A
ij
i j
A
ij
j
[−1; 1]
A
k
[−
1
k
;
1
k
]
k ∈ N
1)
∞
[
k=1
A
k
; 2)
∞
\
k=1
A
k
; 3)
∞
[
k=1
A
c
k
; 4) A
1
∩
∞
\
k=2
A
c
k
.
A
jk
j
k j ∈ N k = 1, 2, 3, 4, 5, 6
A
jk
A
B
C
1) A
13
A
25
A
34
A
46
; 2)
6
\
k=1
A
kk
; 3) A
11
A
25
A
32
A
44
A
53
A
63
.
A B
max[
P
(A),
P
(B)] 6
P
(A ∪ B) 6 2 max[
P
(A),
P
(B)].
P
(A ∪
B) 6
P
(A) +
P
(B) 6 max[
P
(A),
P
(B)] + max[
P
(A),
P
(B)] =
2 max[
P
(A),
P
(B)]
A ⊂ A ∪ B B ⊂ A ∪ B
P
(A) 6
P
(A ∪ B)
P
(B) 6
P
(A ∪ B)
max[
P
(A),
P
(B)] 6
P
(A ∪ B)
A B
P
(A4B) =
P
(A) +
P
(B) − 2
P
(AB).
A B C
P
(A ∪B ∪C) =
P
(A) +
P
(B) +
P
(C) −
P
(AB) −
P
(AC) −
P
(BC) +
P
(ABC).
A B C
P
(A4B) 6
P
(A4C) +
P
(C4B)
A B C
P
(AB) +
P
(AC) +
P
(BC) >
P
(A) +
P
(B) +
P
(C) − 1
P
(AB) +
P
(AC) −
P
(BC) <
P
(A)
A B
P
(AB) =
P
(A
c
B
c
)
A
c
B
c
= (A ∪ B)
c
P
(A
c
B
c
) = 1−
P
(A∪B) = 1−(
P
(A)+
P
(B)−
P
(A∩B)
P
(A) =
P
(B) = 0.5
P
(AB) =
P
(A
c
B
c
)
P
(A) = a
P
(B) = b
P
(C) = c
P
(A ∩ B) = p
P
(A ∩ C) = q
P
(B ∩ C) = r
P
(A ∩ B ∩ C) = h
P
((A4B) \C)
P
(A
c
∩B
c
∩C
c
)
P
((A ∪B) ∩C
c
)
P
(lim sup
n→∞
A
n
) = lim
n→∞
P
∞
[
k=n
A
k
!
,
P
(lim inf
n→∞
A
n
) = lim
n→∞
P
∞
\
k=n
A
k
!
.
lim sup
n→∞
A
n
=
∞
T
n=1
∞
S
k=n
A
k
B
n
=
∞
S
k=n
A
k
n =
1, 2, . . .
B
n
⊃ B
n+1
n lim sup
n→∞
A
n
=
∞
T
n=1
B
n
B
n
lim
n→∞
P
(B
n
) =
P
(
∞
T
n=1
B
n
)
B
n
A
k
P
(lim sup
n→∞
A
n
) =
lim
n→∞
P
∞
S
k=n
A
k
A
n
P
(lim inf
n→∞
A
n
) 6 lim inf
n→∞
P
(A
n
) 6 lim sup
n→∞
P
(A
n
) 6
P
(lim sup
n→∞
A
n
).
P
∞
S
n=1
A
n
=
P
(A
1
) +
P
(A
c
1
A
2
) +
P
(A
c
1
A
c
2
A
3
) + . . . .
A
1
S
A
c
1
A
2
S
A
c
1
A
c
2
A
3
S
··· =
∞
S
n=1
A
n
A
1
A
c
1
A
2
A
c
1
A
c
2
A
3
. . .
P
∞
S
n=1
A
n
=
P
(A
1
)+
P
(A
c
1
A
2
) +
P
(A
c
1
A
c
2
A
3
) + . . . .
P
∞
S
n=1
A
n
= 0
P
(A
n
) = 0 ∀ n
P
∞
T
n=1
A
n
= 1
P
(A
n
) = 1 ∀ n
Ω
R
n
F σ Ω µ
(Ω, F) µ(·) : F 7→ R
+
(A
n
)
n∈N
⊂ F A
i
∩ A
j
= ∅ i 6= j
µ(
∞
[
n=1
A
n
) =
∞
X
n=1
µ(A
n
);
µ(Ω) < ∞
Ω
A ∈ F B ∈ F
µ(A) = µ(B) A
B
(Ω, F,
P
(·))
P
(A) =
µ(A)
µ(Ω)
, A ∈ F. (4.1)
µ
[−1, 1]
p q
x
2
+px+q = 0
p
q
Ω = [−1, 1] × [−1, 1] = {(p, q) : −1 6 p, q 6 1}
D = p
2
− 4q > 0. Ω A
P (A) =
µ(A)
µ(Ω)
µ
µ(Ω) = 4 µ(A)
µ(A) = 2 +
1
Z
−1
p
2
4
dp =
13
6
.
P (A) =
13
24
d
[0, l]
X Y
[0, T ] X
Y t
Y X s
L 5L
R = 2L
{(x, y) : x
2
+ y
2
= R
2
}
r r < R
Ω
A
B
1
B
2
r
OX
A
1
A
2
A
3
A
4
^
A
1
A
2
^
A
3
A
4
OX
^
A
1
A
2
^
A
3
A
4
P (A) =
µ(A)
µ(Ω)
µ
µ(Ω) = 2πR
µ(A) α = ∠OA
1
B
1
4OB
1
A
1
α = arcsin
r
R
4OB
1
A
1
= 4OB
1
A
3
= 4OB
2
A
2
= 4OB
2
A
4
µ(
^
A
1
A
2
) = µ(
^
A
3
A
4
) = 2αR
P (A) =
4 arcsin
r
R
2πR
=
2
π
arcsin
r
R
R
r (r < 2R)
r
√
3r.
R
A 2R
A
r
√
3r
x
2
+ y
2
= R
2
A
R A
r
√
3r
x
2
+ (y −1)
2
= 1
A A
O
0
B O
0
A
P
{|OB| 6
√
3} O
A B C
ABC
K = {(x, y) : |x| < a, |y| < a}
A
A r (r < a) K
K
A r
r K
1
= {(x, y) : |x| <
a − r, |y| < a − r} K
1
P (K
1
) =
µ(K
1
)
µ(K)
=
4(a−r)
2
4a
2
= (1−
r
a
)
2
µ(·)
K = {(x, y)| |x| 6 2, |y| 6 2}
A(x, y) min{|x|, |y|} 6
1
K = {(x, y)| |x| 6 1, |y| 6 1}
A(x, y)
y = x
K = {(x, y)| |x| 6 2, |y| 6 2}
A(x, y)
max{|x|, |y|} 6 1
K = {(x, y)| |x| 6 2, |y| 6 2}
A(x, y) min{|x|, |y|} >
1
K = {(x, y)| |x| 6 2, |y| 6 2}
A(x, y)
max{|x|, |y|} > 1
K = {(x, y)| |x| 6 2, |y| 6 2}
A(x, y)
max{|x|, |y|} 6 min{|x|, |y|}
K = {(x, y)|
x
2
4
+ y
2
= 1}
A(x, y)
OX
1
Ω