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现在我们有诸$x_i$的上界,但它们的处理方法与下界的处理方法不同,我们将在了解\textit{容斥原理}后再解决这个问题。 \subsection{有限概率} - 有限概率的背景是这样的,有一个实验$\epsilon$,在进行这个实验时,它产生的结果是某有限结果集合中的一个。假设每一个结果都是\textbf{等可能的}($equally likely$)(即没有哪一个结果比其他结果更有可能出现。)这时我们说这个实验是\textbf{随机的}($randomly$)。所有可能结果的集合被称为这个实验的\textbf{样本空间}($sample space$),并把它记作$S$ + 有限概率的背景是这样的,有一个实验$\epsilon$,在进行这个实验时,它产生的结果是某有限结果集合中的一个。假设每一个结果都是\textbf{等可能的}($equally likely$)(即没有哪一个结果比其他结果更有可能出现。)这时我们说这个实验是\textbf{随机的}($randomly$)。所有可能结果的集合被称为这个实验的\textbf{样本空间}($sample space$),并把它记作$S$。因此,$S$是一个有限集合,比如说有下面$n$个元素的集合: + \[S = \{s_1 , s_2 , \cdots ,s_n\}\] + 当我们进行实验$\epsilon$时,每有一个$s_i$都有$n$分之一的出现机会,所以说结果$s_i$的概率是$1/n$,写作 + \[Prob(s_i) = 1/n (i = 1,2,\cdots,n)\] + 一个\textbf{事件}($event$) 就是样本空间的一个子集$E$,但是我们通常是用描述式的语言给出这个$E$。 + \paragraph{概率} 在样本空间为$S$的实验中,事件$E$的\textbf{概率}($probability$)定义为$S$中属于$E$的结果的比率,因此, + \[Prob(E) = \frac{|E|}{|S|}\] + 根据定义,事件$E$满足 + \[ 0 \leqslant Prob(E) \leqslant 1\] + 其中,当$Prob(E) = 0$当且仅当$E$是一个空事件$\varnothing$(即不可能事件),而$Prob(E) =1$当且仅当$E$是整个样本空间$E$(即肯定出现的事件)。 \section{鸽巢原理} + \section{附录A} \subsection{数学归纳法正确性的证明} \footnote{本小节参考了百度百科和维基百科数学归纳法词条、皮亚诺公理词条、朱塞佩·皮亚诺词条} diff --git a/note.toc b/note.toc index 38e0924..6d4e646 100644 --- a/note.toc +++ b/note.toc @@ -39,20 +39,21 @@ \contentsline {paragraph}{例子}{10}{paragraph*.20} \contentsline {paragraph}{有限重复数的多重集合的组合}{10}{paragraph*.21} \contentsline {subsection}{\numberline {2.6}有限概率}{11}{subsection.2.6} +\contentsline {paragraph}{概率}{11}{paragraph*.22} \contentsline {section}{\numberline {3}鸽巢原理}{11}{section.3} \contentsline {section}{\numberline {4}附录A}{11}{section.4} \contentsline {subsection}{\numberline {4.1}数学归纳法正确性的证明}{11}{subsection.4.1} -\contentsline {paragraph}{皮亚诺公理(自然数公理)}{11}{paragraph*.22} -\contentsline {subsection}{\numberline {4.2}排列与组合}{11}{subsection.4.2} -\contentsline {subsubsection}{\numberline {4.2.1}结论“乘法原理是加法原理的推论”的证明}{11}{subsubsection.4.2.1} +\contentsline {paragraph}{皮亚诺公理(自然数公理)}{11}{paragraph*.23} +\contentsline {subsection}{\numberline {4.2}排列与组合}{12}{subsection.4.2} +\contentsline {subsubsection}{\numberline {4.2.1}结论“乘法原理是加法原理的推论”的证明}{12}{subsubsection.4.2.1} \contentsline {subsubsection}{\numberline {4.2.2}补集}{12}{subsubsection.4.2.2} -\contentsline {paragraph}{相对补集}{12}{paragraph*.23} -\contentsline {paragraph}{绝对补集}{12}{paragraph*.24} +\contentsline {paragraph}{相对补集}{12}{paragraph*.24} +\contentsline {paragraph}{绝对补集}{12}{paragraph*.25} \contentsline {subsubsection}{\numberline {4.2.3}2.2.1排列公式的证明}{12}{subsubsection.4.2.3} \contentsline {subsubsection}{\numberline {4.2.4}2.2.2循环排列公式的证明}{12}{subsubsection.4.2.4} -\contentsline {subsubsection}{\numberline {4.2.5}2.3.1 集合的组合公式的证明}{12}{subsubsection.4.2.5} -\contentsline {subsubsection}{\numberline {4.2.6}2.4.1 无限重复数的多重集合的排列公式的证明}{12}{subsubsection.4.2.6} +\contentsline {subsubsection}{\numberline {4.2.5}2.3.1 集合的组合公式的证明}{13}{subsubsection.4.2.5} +\contentsline {subsubsection}{\numberline {4.2.6}2.4.1 无限重复数的多重集合的排列公式的证明}{13}{subsubsection.4.2.6} \contentsline {subsubsection}{\numberline {4.2.7}2.4.2 有限重复数的多重集合的排列公式证明}{13}{subsubsection.4.2.7} \contentsline {subsection}{\numberline {4.3}其他}{13}{subsection.4.3} \contentsline {subsubsection}{\numberline {4.3.1}算术基本定理}{13}{subsubsection.4.3.1} -\contentsline {paragraph}{算术基本定理}{13}{paragraph*.25} +\contentsline {paragraph}{算术基本定理}{14}{paragraph*.26}