概率论【大学常用公式】
贝叶斯公式:

\begin{array}{*{20}{l}}
{\text{若}\text{有}\mathop{ \bigcup }\limits_{{k=1}}^{{n}}\mathop{{B}}\nolimits_{{k}}=S}\\
{\text{且}\text{有}\mathop{{B}}\nolimits_{{i}}\mathop{{B}}\nolimits_{{j}}= \emptyset { \left( {i \neq j,i,j=1,2,3, \cdots n} \right) }}\\
{\text{且}\text{有}P{ \left( {\mathop{{B}}\nolimits_{{k}}} \right) } > 0{ \left( {k=1,2,3, \cdots n} \right) }}\\
{\text{且}\text{有}P{ \left( {A} \right) } > 0}\\
{\text{则}\text{有}P{ \left( {\mathop{{B}}\nolimits_{{k}} \left| A\right. } \right) }=\frac{{P \left( {\mathop{{B}}\nolimits_{{k}}} \left) \cdot P{ \left( {A \left| {\mathop{{B}}\nolimits_{{k}}}\right. } \right) }\right. \right. }}{{\mathop{ \sum }\limits_{{i=1}}^{{n}}P{ \left( {\mathop{{B}}\nolimits_{{i}}} \right) } \cdot P{ \left( {A \left| \mathop{{B}}\nolimits_{{i}}\right. } \right) }}}}
\end{array}

超几何概率分布:

\begin{array}{*{20}{l}}
{S={ \left( {\begin{array}{*{20}{c}}
{N}\\
{n}
\end{array}} \right) },\mathop{{A}}\nolimits_{{k}}= \left( {\begin{array}{*{20}{c}}
{M}\\
{k}
\end{array}} \left) \cdot \left( {\begin{array}{*{20}{c}}
{N-M}\\
{n-k}
\end{array}} \right) \right. \right. }\\
{P{ \left( {\mathop{{A}}\nolimits_{{k}}} \right) }=\frac{{ \left( {\begin{array}{*{20}{c}}
{M}\\
{k}
\end{array}} \left) \cdot \left( {\begin{array}{*{20}{c}}
{N-M}\\
{n-k}
\end{array}} \right) \right. \right. }}{{ \left( {\begin{array}{*{20}{c}}
{N}\\
{n}
\end{array}} \right) }}}
\end{array}

独立性定理:

\begin{array}{*{20}{l}}
{\text{若}P{ \left( {AB} \right) }=P{ \left( {A} \right) }P{ \left( {B} \right) }}\\
{\text{则}P{ \left( {A \left| B\right. } \right) }=P{ \left( {B} \right) }}
\end{array}

概率的乘法公式:

\begin{array}{*{20}{l}}
{P{ \left( {AB} \right) }=P{ \left( {A} \right) }P{ \left( {B \left| A\right. } \right) }}\\
{P{ \left( {ABC} \right) }=P{ \left( {A} \right) }P{ \left( {B \left| A\right. } \right) }P{ \left( {C \left| AB\right. } \right) }}
\end{array}

概率的基本性质1:

\begin{array}{*{20}{l}}
{P{ \left( { \emptyset } \right) }=0}\\
{P{ \left( {S} \right) }=1}
\end{array}

概率的基本性质2:

\begin{array}{*{20}{l}}
{ \forall A \in S}\\
{P{ \left( {A} \right) } \ge 0}
\end{array}

概率的基本性质3:

\begin{array}{*{20}{l}}
{\text{若}\mathop{{A}}\nolimits_{{i}}\mathop{{A}}\nolimits_{{j}}= \emptyset { \left( {i \neq j,i,j={1,2,3, \cdots }} \right) },\text{则}\text{有}}\\
{P{ \left( {\mathop{{A}}\nolimits_{{1}} \bigcup \mathop{{A}}\nolimits_{{2}} \bigcup \cdots \bigcup \mathop{{A}}\nolimits_{{n}} \cdots } \right) }=P{ \left( {\mathop{{A}}\nolimits_{{1}}} \right) }+P{ \left( {\mathop{{A}}\nolimits_{{2}}} \right) }+ \cdots P{ \left( {\mathop{{A}}\nolimits_{{n}}} \right) }+ \cdots }\\
{\text{或}\text{者}}\\
{P{ \left( {\mathop{ \bigcup }\limits_{{i=1}}^{{+ \infty }}\mathop{{A}}\nolimits_{{i}}} \right) }=\mathop{ \prod }\limits_{{i=1}}^{{+ \infty }}P{ \left( {\mathop{{A}}\nolimits_{{i}}} \right) }}
\end{array}

概率性质1:

\begin{array}{*{20}{l}}
{\text{若}\mathop{{A}}\nolimits_{{i}}\mathop{{A}}\nolimits_{{j}}= \emptyset { \left( {i \neq j,i,j={1,2,3, \cdots n}} \right) },\text{则}\text{有}}\\
{P{ \left( {\mathop{{A}}\nolimits_{{1}} \bigcup \mathop{{A}}\nolimits_{{2}} \bigcup \cdots \bigcup \mathop{{A}}\nolimits_{{n}}} \right) }=P{ \left( {\mathop{{A}}\nolimits_{{1}}} \right) }+P{ \left( {\mathop{{A}}\nolimits_{{2}}} \right) }+ \cdots P{ \left( {\mathop{{A}}\nolimits_{{n}}} \right) }}\\
{\text{或}\text{者}}\\
{P{ \left( {\mathop{ \bigcup }\limits_{{i=1}}^{{n}}\mathop{{A}}\nolimits_{{i}}} \right) }=\mathop{ \prod }\limits_{{i=1}}^{{n}}P{ \left( {\mathop{{A}}\nolimits_{{i}}} \right) }}
\end{array}

概率性质2:

\begin{array}{*{20}{l}}
{ \forall A,B \in S}\\
{P{ \left( {A-B} \right) }=P{ \left( {A} \right) }-P{ \left( {AB} \right) }}
\end{array}

概率性质3:

\begin{array}{*{20}{l}}
{ \forall A \in S}\\
{P{ \left( { \overline {A}} \right) }=1-P{ \left( {A} \right) }}
\end{array}

概率性质4:

\begin{array}{*{20}{l}}
{ \forall A,B \in S}\\
{P{ \left( {A \bigcup B} \right) }=P{ \left( {A} \left) +P{ \left( {B} \right) }\right. \right. }-P{ \left( {AB} \right) }}
\end{array}

概率与频率的关系:

$$P{ \left( {A} \right) }={\mathop{{\text{l}\text{i}\text{m}}}\limits_{{n \to \infty }}}{\mathop{{f}}\nolimits_{{n}}{ \left( {A} \right) }}$$

古典概率:

\begin{array}{*{20}{l}}
{S={ \left\{ {\mathop{{e}}\nolimits_{{1}},\mathop{{e}}\nolimits_{{2}},\mathop{{e}}\nolimits_{{3}}, \cdots \mathop{{e}}\nolimits_{{n}}} \right\} }}\\
{P{{ \left( { \left\{ {\mathop{{e}}\nolimits_{{1}}} \right\} } \right) }}=P{ \left( { \left\{ {\mathop{{e}}\nolimits_{{2}}} \right\} } \right) }=P{ \left( { \left\{ {\mathop{{e}}\nolimits_{{3}}} \right\} } \right) }= \cdots =P{ \left( { \left\{ {\mathop{{e}}\nolimits_{{n}}} \right\} } \right) }}
\end{array}

频率的定义:

$$\mathop{{f}}\nolimits_{{n}}{ \left( {A} \right) }=\frac{{\mathop{{n}}\nolimits_{{A}}}}{{n}}$$

频率性质1:

$$0 \le \mathop{{f}}\nolimits_{{n}}{ \left( {A} \right) } \le 1$$

频率性质2:

$$\mathop{{f}}\nolimits_{{n}}{ \left( {S} \right) }=1$$

频率性质3:

\begin{array}{*{20}{c}}
{A \bigcap B= \emptyset }\\
{ \Rightarrow \mathop{{f}}\nolimits_{{n}}{ \left( {A+B} \right) }=\mathop{{f}}\nolimits_{{n}}{ \left( {A} \right) }+\mathop{{f}}\nolimits_{{n}}{ \left( {B} \right) }}
\end{array}

全概率公式:

\begin{array}{*{20}{l}}
{\text{若}\text{有}\mathop{ \bigcup }\limits_{{k=1}}^{{n}}\mathop{{B}}\nolimits_{{k}}=S}\\
{\text{且}\text{有}\mathop{{B}}\nolimits_{{i}}\mathop{{B}}\nolimits_{{j}}= \emptyset { \left( {i \neq j,i,j=1,2,3, \cdots n} \right) }}\\
{\text{且}\text{有}P{ \left( {\mathop{{B}}\nolimits_{{k}}} \right) } > 0{ \left( {k=1,2,3, \cdots n} \right) }}\\
{\text{则}\text{有}P{ \left( {A} \right) }=\mathop{ \sum }\limits_{{k=1}}^{{n}}P{ \left( {\mathop{{B}}\nolimits_{{k}}} \right) } \cdot P{ \left( {A \left| \mathop{{B}}\nolimits_{{k}}\right. } \right) }}
\end{array}

三事件互相独立:

$${\begin{array}{*{20}{l}}
{P{ \left( {AB} \right) }=P{ \left( {A} \right) }P{ \left( {B} \right) }}\\
{P{ \left( {AC} \right) }=P{ \left( {A} \right) }P{ \left( {C} \right) }}\\
{P{ \left( BC \right) }=P{ \left( B \right) }P{ \left( C \right) }}
\end{array}} \left\} \Rightarrow P{ \left( {ABC} \right) }=P{ \left( {A} \right) }P{ \left( {B} \right) }P{ \left( {C} \right) }\right.$$

条件概率:

\begin{array}{*{20}{l}}
{\text{定}\text{义}\text{在}\text{事}\text{件}B\text{已}\text{经}\text{发}\text{生}\text{情}\text{况}\text{下}\text{,}}\\
{\text{事}\text{件}A\text{发}\text{生}\text{的}\text{条}\text{件}\text{概}\text{率}\text{为}}\\
{P{ \left( {A \left| B\right. } \right) }=\frac{{P{ \left( {AB} \right) }}}{{P{ \left( {B} \right) }}}}\\
{\text{其}\text{中}P{ \left( {B} \right) } > 0}
\end{array}

 

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