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120 words
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1.3 条件概率
2025-11-26
Probability Theory
/
math
/
probability theory
/
notes
1.3 条件概率
#
定义
#
条件概率
（
P
(
B
)
>
0
P(B) > 0
P
(
B
)
>
0
）：
P
(
A
∣
B
)
=
P
(
A
∩
B
)
P
(
B
)
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
P
(
A
∣
B
)
=
P
(
B
)
P
(
A
∩
B
)
等可能模型：
P
(
A
∣
B
)
=
事件
A
∩
B
的结果数
事件
B
的结果数
P(A \mid B) = \frac{\text{事件}A \cap B\text{的结果数}}{\text{事件}B\text{的结果数}}
P
(
A
∣
B
)
=
事件
B
的结果数
事件
A
∩
B
的结果数
性质
#
P
(
⋅
∣
B
)
P(\cdot \mid B)
P
(
⋅
∣
B
)
是合格的概率律
满足非负性、归一化、可加性
可将
B
B
B
视为新的样本空间
乘法规则
#
两个事件
：
P
(
A
∩
B
)
=
P
(
A
)
P
(
B
∣
A
)
=
P
(
B
)
P
(
A
∣
B
)
P(A \cap B) = P(A)P(B \mid A) = P(B)P(A \mid B)
P
(
A
∩
B
)
=
P
(
A
)
P
(
B
∣
A
)
=
P
(
B
)
P
(
A
∣
B
)
多个事件
：
P
(
⋂
i
=
1
n
A
i
)
=
P
(
A
1
)
P
(
A
2
∣
A
1
)
P
(
A
3
∣
A
1
∩
A
2
)
⋯
P
(
A
n
∣
⋂
i
=
1
n
−
1
A
i
)
P(\bigcap_{i=1}^n A_i) = P(A_1)P(A_2 \mid A_1)P(A_3 \mid A_1 \cap A_2) \cdots P(A_n \mid \bigcap_{i=1}^{n-1} A_i)
P
(
⋂
i
=
1
n
A
i
)
=
P
(
A
1
)
P
(
A
2
∣
A
1
)
P
(
A
3
∣
A
1
∩
A
2
)
⋯
P
(
A
n
∣
⋂
i
=
1
n
−
1
A
i
)
1.3 条件概率
https://miku.nikonikoni.blog/posts/propability_theory/1-3-conditional-probability/
Author
nikonikoni
Published at
2025-11-26
License
Unlicensed
Some information may be outdated
1.2 概率模型
1.4 全概率定理和贝叶斯准则
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