Дронов С.В. Задачник по теории вероятностей (первый семестр)
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(ξ, η)
x
2
+ ξx + η = 0
ξx
2
+ ηx + ζ = 0,
(ξ, η, ζ)
d
R
R
√
3.
2a
2l (l < a).
a
a
d
R
n
a
2L
a
v u
P(A ∪ B) = P(A) + P(B) −P(AB)
P(AB) = P(A/B) ·P(B).
A, B
P(A ∪ B) = P(A) + P(B).
σ
P(∪
∞
n=1
A
n
) =
∞
X
n=1
P(A
n
).
A, B
P(AB) = P(A) · P(B).
B
A
B C = A ∪ B
P(A) = P(B) = 1/6, A, B
P(AB) =
1
6
×
1
6
=
1
36
P(C) = P(A) + P(B) − P(A)P(B) =
11
36
.
¯
C =
¯
A
¯
B. P(
¯
A) = P(
¯
B) =
5
6
,
P(
¯
C) = P(
¯
A)P(
¯
B) =
25
36
; P(C) = 1 − P(
¯
C) =
11
36
.
j
j = 1, 2, ...
A
j
= { j };
B
j
= { j },
C = { }; D = { }.
C = A
1
∪ (∪
∞
k=1
(B
1
...B
2k
A
2k+1
);
D = ∪
∞
k=1
(B
1
...B
2k−1
A
2k
),
P(C) = P(A
1
) +
P
∞
k=1
P(B
1
)...P(B
2k
)P(A
2k+1
);
P(D) =
P
∞
k=1
P(B
1
)...P(B
2k−1
)P(A
2k
).
P(A
j
) = 3/10, P(B
k
) = 7/10 j k
P(C) =
3
10
×
1 +
P
∞
j=1
7
10
j
=
10
17
;
P(D) =
3
10
×
P
∞
j=1
7
10
j
=
7
17
.
P(A
1
...A
n
) = P(A
n
/A
1
...A
n−1
) × P(A
n−1
/A
1
...A
n−2
) × ...
×P(A
3
/A
1
A
2
)P(A
2
/A
1
)P(A
1
),
P(∪
n
j=1
A
j
) =
P
j
P(A
j
) −
P
i<j
P(A
i
A
j
) +
P
i<j<k
P(A
i
A
j
A
k
) − ...
+ (−1)
n−1
P(A
1
...A
n
).
n
P(A
1
...A
k
A
k+1
) = P(A
k+1
/A
1
...A
k
) × P(A
1
...A
k
)
P(A
1
...A
k
)
n = k + 1
P((∪
n
j=1
A
j
) ∪ A
n+1
) = P(∪
n
j=1
A
j
) + P(A
n+1
) − P(∪
n
j=1
A
j
A
n+1
).
A
1
, ..., A
n
A
1
A
n+1
, ..., A
n
A
n+1
P(A
j
), j = 1, ., n + 1
A
n+1
n
2
p
i,j
, i, j = 1, ..., n
k
P(A) =
n
X
k=1
P(A/H
k
)P(H
k
).
H
i
∪
n
k=1
H
k
= Ω
i 6= j ⇒ H
i
∩ H
j
= ∅
j P(H
j
) 6= 0.
P(A) H
i
P(H
1
) = P(H
2
) = ... = P(H
n
) =
1
n
.
P(H
j
/A) =
P(A/H
j
)P(H
j
)
P
n
k=1
P(A/H
k
)P(H
k
)
. j = 1, ..., n.
A = { };
H
1
= { };
H
2
= { }.
P(H
1
) =
5
25
=
1
5
; P(H
2
) =
20
25
=
4
5
;
P(A/H
1
) =
4
24
=
1
6
; P(A/H
2
) =
5
24
,
P(A) =
1
5
×
1
6
+
4
5
×
5
24
=
1
5
.
A = { },
H
i
= { i }, i = 1, 2, 3.
P(H
1
) = 0, 45, P(H
2
) = 0, 35, P(H
3
) = 0, 2,
P(A/H
1
) = 0, 02, P(A/H
2
) = 0, 03, P(A/H
3
) = 0, 05,
P(A) = 0, 0295.
P(H
1
/A) = 0, 45 × 0, 02/0, 0295 ≈ 0, 3051,
P(H
2
/A) = 0, 35 × 0, 03/0, 0295 ≈ 0, 3559,
P(H
3
/A) = 0, 20 × 0, 05/0, 0295 ≈ 0, 3390.
n
n
k
p 1 − p
H
1
, ..., H
n
ξ x
k
, k =
1, 2, ...
p
k
= P(ξ = x
k
), p
i,k
= P(ξ = x
k
/H
i
), k = 1, 2..., i = 1, ..., n.
Mξ =
X
k
x
k
p
k
; M(ξ/H
i
) =
X
k
x
k
p
i,k
.
Mξ =
n
X
k=1
M(ξ/H
k
)P(H
k
).
p
P
n
(k) k n
P
n
(k) = C
k
n
p
k
q
n−k
, q = 1 − p.
n p k
P
n
(k) ≈
(np)
k
k!
e
−np
.
0 < np < 9
n → ∞, k−np → ∞,
P
n
(k) ≈
1
√
npq
ϕ
k − np
√
npq
!
, ϕ(x) =
1
√
2π
e
−x
2
/2
−
P(s ≤ k ≤ m) =
m
X
k=s
P
n
(k)