Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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µ(Ω) = 6
µ
A
C
C
C 1 µ(A) =
3π
4
µ
P (A) =
µ(A)
µ(Ω)
=
π
8
π
3
1
2
60
◦
60
◦
30
◦
AB l
L M L
M
AB l
M N
a
l
3
6 a 6 l
20
1
A B
P
(A|B) =
P
(A ∩ B)
P
(B)
.
A
J (H
j
)
j∈J
P
(H
j
) 6= 0 j ∈ J H
j
∩H
i
= ∅ i 6= j A ⊂
S
j∈J
H
j
P
(A) =
X
j∈J
P
(A|H
j
)
P
(H
j
).
P
(A) 6= 0
P
(H
k
|A) =
P
(A|H
k
)
P
(H
k
)
P
j∈J
P
(A|H
j
)
P
(H
j
)
, k ∈ J.
A B
P
(A ∩ B) =
P
(A)
P
(B)
A
1
, . . . , A
n
P
(A
j
1
∩ . . . ∩ A
j
k
) =
P
(A
j
1
) · . . . ·
P
(A
j
k
)
k 1 < k 6 n j
1
, . . . , j
k
1 6 j
1
< . . . < j
k
6 n
A
1
, A
2
, . . .
n > 1 A
1
, . . . , A
n
A
1
, . . . , A
n
B
j
= A
j
B
j
= A
c
j
j = 1, . . . , n
B
1
, . . . , B
n
A
c
B
A
c
∩ B
P
(B) =
3
3
6
3
=
1
8
P
(A
c
∩ B) =
2
3
6
3
=
1
27
P
(A
c
|B) =
P
(A
c
∩B)
P
(B)
=
8
27
P
(A|B) =
1 −
P
(A
c
|B) =
19
27
3, 7
α β γ
A B
A B p q A
A
i
A
i B
j
B j A
A B
B A = A
1
∪ A
c
1
B
c
1
A
2
∪ . . .
B = A
c
1
B
1
∪ A
c
1
B
c
1
A
c
2
B
2
∪ . . .
A
1
, B
1
, A
2
, B
2
. . .
P
(A) = p + (1 − p)(1 − q)p + (1 − p)(1 − q)(1 − p)(1 − q)p + ··· =
= p(1 + (1 − p)(1 − q) + (1 − p)
2
(1 − q)
2
+ ···) =
p
1 − (1 − p)(1 − q)
.
P
(B) = (1−p)q+(1−p)(1−q)(1−p)q+(1−p)(1−q)(1−p)(1−q)(1−p)q+···
= (1 −p)q(1 + (1 −p)(1 −q) + (1 −p)
2
(1 −q)
2
+ ···) =
(1 − p)q
1 − (1 − p)(1 − q)
.
P
(A) +
P
(B) =
p+(1−q)p
1−(1−p)(1−q)
=
p+q−pq
p+q−pq
= 1
0
a b
3 5 2
1
A
1
A
2
B
3 A
1
2
A
B
1
2
n
n
A
1
, . . . , A
n
P
(A
k
) = p
k
A
1
, . . . , A
n
A
1
, . . . , A
n
A
1
, . . . , A
n
m
p
n
R N
r (0 < r < R)
n
H
k
k k = 0, n
P
(H
k
) =
1
n+1
k
P
(A) =
n
X
k=0
P
(A|H
k
)
P
(H
k
).
P
(A|H
k
)
k
P
(A|H
k
) =
k
n
.
P
(A) =
n
X
k=0
k
n
1
n + 1
=
1
n(n + 1)
n
X
k=1
k =
1
n(n + 1)
(1 + n)n
2
=
1
2
.
n
n
1
n
2
k
1
k
2
n
1
m
1
n
2
m
2
k
1
k
2
p
1
p
2
p
3
N M
A
H
1
H
2
P
(H
1
) =
N
N+M
P
(H
2
) =
M
N+M
P
(A|H
i
) i = 1, 2
P
(A|H
1
) =
C
2
N−1
C
2
N+M −1
=
(N − 1)(N −2)
(N + M − 1)(N + M − 2)
,
P
(A|H
2
) =
C
2
N
C
2
N+M −1
=
N(N − 1)
(N + M − 1)(N + M − 2)
.
P
(A) =
(N − 1)(N −2)
(N + M −1)(N + M − 2)
N
N + M
+
+
N(N − 1)
(N + M − 1)(N + M − 2)
M
N + M
=
N(N − 1)
(N + M)(N + M − 1)
.
P
(H
1
|A) =
P
(A|H
1
)
P
(H
1
)
P
(A)
=
(N−1)(N −2)
(N+M −1)(N +M−2)
·
N
N+M
N(N −1)
(N+M )(N +M−1)
=
N − 2
N + M − 2
.
n
k
1
m
1
n
1
k
2
m
2
n
2
p
1
p
2
p
3
1 2 3
n
1
m
1
n
2
m
2
p q
P
(A)
A B
P
(A
T
B) =
P
(A)
P
(B)
B = A
P
(A
T
A) =
P
(A) =
P
2
(A) P (A) = 0
P (A) = 1
A B
P
(A) > 0
P
(B) > 0
A B
A B
P
(A
T
B) =
P
(A)
P
(B)
A B
P
(A
T
B) = 0
P
(A) > 0
P
(B) > 0
P
(A)
P
(B) > 0
A B A C B
C A B ∪ C
A B
P
(A ∪B) = 1
P
(A) = 1
P
(B) = 1
0 <
P
(B) < 1
P
(A | B) =
P
(A |
B
c
) A B
A
P
(A)
0 1
A B
A B
A∪B A∩B
P
(A) = 1
P
(B) =
1
P
(A) = 0
P
(B) = 0
A B
P
(A∪B) =
P
(A)+
P
(B)
P
(A4B) = p
P
(A \ B) < p
P
(A)
P
(B)
P
(A \ B)
A
B
C A
B C
(Ω, F, P )
ξ(·) : Ω 7→ R
{ξ 6 x} ∈ F x ∈ R
X
R ξ
P
{ξ ∈ X} = 1
ξ ξ
p
ξ
(x) ,
P
{ξ = x}, x ∈ X,
ξ
X
R (p(x))
x∈X
(p(x))
x∈X
p(x) > 0 ∀x ∈ X
P
x∈X
p(x) = 1
ξ F
ξ
(x) , P {ξ 6 x} x ∈ R
−∞ < a < b < +∞
F
ξ
(a) 6 F
ξ
(b),
P
{a < ξ 6 b} = F
ξ
(b) − F
ξ
(a);
lim
x→−∞
F
ξ
(x) = 0, lim
x→+∞
F
ξ
(x) = 1
F
ξ
(x)
P
{ξ < x} = F
ξ
(x−) F
ξ
(x−) = lim
y↑x
F
ξ
(y)
F
ξ
x
P
{ξ = x} = F
ξ
(x) − F
ξ
(x−)
ξ
Y
F
ξ
(x) =
P
y∈Y, y6x
P
{ξ = y}
P
{ξ = y}
y ∈ Y
F : R 7→ R
F
lim
x→−∞
F (x) = 0 lim
x→+∞
F (x) = 1
ξ
f
ξ
(·) : R 7→ R
+
F
ξ
(x) =
x
Z
−∞
f
ξ
(u)du x ∈ R,
f
ξ
(x)
ξ