Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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AB A, B, A, B
A + B A, B, A + B
A B A, B, AB
A + B A, B, A + B
A
B
P (A), P (B), P (A|B), P (B|A), P (AB), P (A + B)
2 : 5
1, 2, . . . , k
p 1 − p
1, 2, . . . , k
n
•
•
•
•
A
B A
B
B B
A B
B
A
P (B|A) = P (B).
A
B
P (B|A) =
P (AB)
P (A)
=
1
6
:
1
2
=
1
3
P (B) =
1
3
,
B A
B A
A B
P (AB) = P (A)P (B|A) = P (B)P (A|B).
B A
P (B|A) = P (B).
P (A)P (B) = P (B)P (A|B),
P (B)
P (A) = P (A|B).
A
B ¤
A B
A B
P (AB) = P (A)P (B).
P (AB) = P (A)P (B|A),
A B P (B|A) = P (B)
P (AB) = P (A)P (B) ¤
A B
P (AB) = P (A)P (B)
A B
A B A B A
B
B
B = AB + AB.
P (B) = P (AB) + P (AB),
P (AB) = P (B) − P (AB).
A B
P (AB) = P (B) − P (A)P (B) = P (B)(1 − P (A)) = P (B)P (A),
P (AB) = P (A)P (B)
A B A B ¤
A
1
, A
2
, . . . , A
n
A
i
1
, A
i
2
, . . . , A
i
k
P (A
i
1
A
i
2
···A
i
k
) = P (A
i
1
)P (A
i
2
) ···P (A
i
k
).
P (A
1
A
2
···A
n
) = P (A
1
)P (A
2
) ···P (A
n
).
A
1
A
2
B
B
B = A
1
A
2
+ A
1
A
2
.
A
1
A
2
A
1
A
2
A
1
A
2
P (B) = P (A
1
A
2
+ A
1
A
2
) =
= P (A
1
A
2
) + P (A
1
A
2
) = P (A
1
)P (A
2
) + P (A
1
)P (A
2
) =
=
4
10
·
7
10
+
6
10
·
3
10
= 0, 46. ¤
p
1
, p
2
, p
3
A
1
A
2
A
3
B
B
B = A
1
A
2
A
3
+ A
1
A
2
A
3
+ A
1
A
2
A
3
.
P (A
1
) = p
1
P (A
2
) = p
2
P (A
3
) = p
3
P (B) = p
1
p
2
(1−p
3
)+p
1
(1−p
2
)p
3
+(1−p
1
)p
2
p
3
. ¤
A
i
i
ω
1
= A
1
A
2
A
3
A
4
, ω
2
= A
1
A
2
A
3
A
4
, . . . , ω
16
= A
1
A
2
A
3
A
4
,
P (A
i
) =
1
6
, P (A
i
) =
5
6
,
P (ω
1
) =
µ
1
6
¶
4
, P (ω
2
) =
µ
1
6
¶
3
·
5
6
, . . . , P (ω
16
) =
µ
5
6
¶
4
.
B
B
ω
4
, ω
6
, ω
7
, ω
10
, ω
11
, ω
13
,
µ
1
6
¶
2
µ
5
6
¶
2
P (B) = 6 ·
µ
1
6
¶
2
µ
5
6
¶
2
¤
n
n
n
n
A
n
A k
A
p
n
(x
1
, x
2
, . . . , x
n
)
k = 1, n
x
i
=
1, i A
,
0, i A
Ω
n
A
A
A Ω
n
n
Ω
A
n
(2)
= 2
n
.
ω = (x
1
, x
2
, . . . , x
n
)
n
A
(x
i
)
i
=
½
A
(1)
i
, i A ,
A
(0)
i
, i A
A
(x
1
)
1
, A
(x
2
)
2
, . . . , A
(x
n
)
n
ω
ω = A
(x
1
)
1
A
(x
2
)
2
···A
(x
n
)
n
,
P (ω) = P (A
(x
1
)
1
)P (A
(x
2
)
2
) ···P (A
(x
n
)
n
) = p
µ(ω)
(1 − p)
n−µ(ω)
,