Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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A
1
A
2
A
3
P (A) = P (A
1
) + P (A
2
) + P (A
3
).
C
3
100
5 · C
2
95
A
1
C
2
5
· C
1
95
A
2
C
3
5
A
3
P (A) =
5 · C
2
95
C
3
100
+
C
2
5
· 95
C
3
100
+
C
3
5
C
3
100
≈
≈ 0, 13806 + 0, 00588 + 0, 0006 = 0, 14400. ¤
A
P (A) = 1 − P (A) = 1 −
C
3
95
C
3
100
≈ 1 − 0, 85600 = 0, 144. ¤
A
B
A + B
A B
A B AB
10 · 10
A
B
AB
P (A) =
60
100
= 0, 6; P (B) =
40
100
= 0, 4; P (AB) =
24
100
= 0, 24.
P (A+B) = 0, 6+0, 4−0, 24 = 0, 76. ¤
n
m A k
B l
n = 20 m = 9 k = 11 l = 4
P (A) =
m
n
.
B A
k
B l A
A B
l
k
P (B) > 0
A B
A B B
P (A|B) =
P (AB)
P (B)
.
P (A) > 0
B A
P (B|A) =
P (AB)
P (A)
.
P (AB) = P (A)P (B|A) = P (B)P (A|B).
P (A
1
A
2
A
3
) = P (A
1
)P (A
2
|A
1
)P (A
3
|A
1
A
2
).
B
i
i
P (A) = 1 − P (A) = 1 − P (B
1
B
2
B
3
) =
= 1 − P (B
1
)P (B
2
|B
1
)P (B
3
|B
1
B
2
) =
= 1 −
95
100
·
94
99
·
93
98
≈ 0, 144.
A B
A
B B A
BA BA
A
A = BA + BA P (A) = P (BA) + P (BA).
P (A) = P (B)P (A|B) + P (B)P (A|B).
A
B B A
B B
A
A
H
1
, H
2
, . . . , H
n
H
i
H
i
H
j
= ∅ (i 6= j), H
1
+ H
2
+ ··· + H
n
= Ω,
A
A = H
1
A + H
2
A + ··· + H
n
A.
P (A) = P (H
1
)P (A|H
1
) + P (H
2
)P (A|H
2
) + ··· + P (H
n
)P (A|H
n
)
P (A) =
n
X
k=1
P (H
k
)P (A|H
k
).
P (H
1
), P (H
2
), . . . , P (H
n
)
A
A
A H
k
k = 1, n
P (AH
k
) = P (A)P (H
k
|A) = P (H
k
)P (A|H
k
).
P (A) > 0
P (H
k
|A) =
P (H
k
)P (A|H
k
)
P (A)
,
P (A)
A
A
A
H
1
H
2
H
3
P (H
1
) =
C
2
25
C
2
30
=
20
29
, P (H
2
) =
25 · 5
C
2
30
=
25
87
, P (H
3
)
C
2
5
C
2
30
=
2
87
.
P (A|H
1
) = 1, P (A|H
2
) =
24
28
=
6
7
, P (A|H
3
) =
C
2
25
C
2
28
=
50
63
.
P (A) = P (H
1
)P (A|H
1
) + P (H
2
)P (A|H
2
) + P (H
3
)P (A|H
3
) =
=
20
29
· 1 +
25
87
·
6
7
+
2
87
·
50
63
≈ 0, 95.
H
1
, H
2
, H
3
P (H
1
) =
C
2
4
C
2
10
=
2
15
, P (H
2
) =
6 · 4
C
2
10
=
8
15
, P (H
3
) =
C
2
6
C
2
10
=
1
3
.
A
P (A|H
1
) =
1
C
2
8
=
1
28
.
P (A|H
2
) =
C
2
3
C
2
8
=
3
28
,
P (A|H
3
) =
C
2
4
C
2
8
=
6
28
.
P (A) =
2
15
·
1
28
+
8
15
·
3
28
+
1
3
·
6
28
=
2
15
.
P (H
1
|A) =
P (H
1
)P (A|H
1
)
P (A)
=
2
15
·
1
28
2
15
=
1
28
,
P (H
2
|A) =
P (H
2
)P (A|H
2
)
P (A)
=
8
15
·
3
28
2
15
=
12
28
,
P (H
3
|A) =
P (H
3
)P (A|H
3
)
P (A)
=
1
3
·
6
28
2
15
=
15
28
.
¤
H
1
H
2
H
3
P (H
1
) = P (H
2
) = P (H
3
) =
1
3
.
3
4
,
1
2
,
1
4
A
P (A|H
1
) =
9
16
, P (A|H
2
) =
4
16
, P (A|H
3
) =
1
16
.
P (A) =
1
3
·
9
16
+
1
3
·
4
16
+
1
3
·
1
16
=
14
48
,
P (H
2
|A) =
1
3
·
4
16
14
16
=
2
7
.
P (H
1
|A) =
1
3
·
9
16
14
48
=
9
14
,
P (H
3
|A) =
1
3
·
1
16
14
48
=
1
4
,
9
14
4
14
1
14
¤
A, B, C
A
A B C
A B
A + B = A · B
AB = A + B
(A + B) + (A + B) = A
(A + B)(A + B)(A + B) = AB
A B
A B
A B
A \ B = A · B
A ⊂ B A \ B = ∅
A B A \ B = A