有限概率

note.aux

@@ -58,20 +58,21 @@
 \@writefile{toc}{\contentsline {paragraph}{例子}{10}{paragraph*.20}}
 \@writefile{toc}{\contentsline {paragraph}{有限重复数的多重集合的组合}{10}{paragraph*.21}}
 \@writefile{toc}{\contentsline {subsection}{\numberline {2.6}有限概率}{11}{subsection.2.6}}
+\@writefile{toc}{\contentsline {paragraph}{概率}{11}{paragraph*.22}}
 \@writefile{toc}{\contentsline {section}{\numberline {3}鸽巢原理}{11}{section.3}}
 \@writefile{toc}{\contentsline {section}{\numberline {4}附录A}{11}{section.4}}
 \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}数学归纳法正确性的证明}{11}{subsection.4.1}}
-\@writefile{toc}{\contentsline {paragraph}{皮亚诺公理（自然数公理）}{11}{paragraph*.22}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}排列与组合}{11}{subsection.4.2}}
-\@writefile{toc}{\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}}
+\@writefile{toc}{\contentsline {paragraph}{皮亚诺公理（自然数公理）}{11}{paragraph*.23}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}排列与组合}{12}{subsection.4.2}}
+\@writefile{toc}{\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}}
 \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.2}补集}{12}{subsubsection.4.2.2}}
-\@writefile{toc}{\contentsline {paragraph}{相对补集}{12}{paragraph*.23}}
-\@writefile{toc}{\contentsline {paragraph}{绝对补集}{12}{paragraph*.24}}
+\@writefile{toc}{\contentsline {paragraph}{相对补集}{12}{paragraph*.24}}
+\@writefile{toc}{\contentsline {paragraph}{绝对补集}{12}{paragraph*.25}}
 \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.3}2.2.1排列公式的证明}{12}{subsubsection.4.2.3}}
 \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.4}2.2.2循环排列公式的证明}{12}{subsubsection.4.2.4}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.5}2.3.1 集合的组合公式的证明}{12}{subsubsection.4.2.5}}
-\@writefile{toc}{\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}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.5}2.3.1 集合的组合公式的证明}{13}{subsubsection.4.2.5}}
+\@writefile{toc}{\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}}
 \@writefile{toc}{\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}}
 \@writefile{toc}{\contentsline {subsection}{\numberline {4.3}其他}{13}{subsection.4.3}}
 \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.3.1}算术基本定理}{13}{subsubsection.4.3.1}}
-\@writefile{toc}{\contentsline {paragraph}{算术基本定理}{13}{paragraph*.25}}
+\@writefile{toc}{\contentsline {paragraph}{算术基本定理}{14}{paragraph*.26}}

note.log

@@ -1,4 +1,4 @@
-This is XeTeX, Version 3.14159265-2.6-0.99998 (TeX Live 2017/Debian) (preloaded format=xelatex 2019.5.2)  4 AUG 2019 17:46
+This is XeTeX, Version 3.14159265-2.6-0.99998 (TeX Live 2017/Debian) (preloaded format=xelatex 2019.5.2)  4 AUG 2019 22:42
 entering extended mode
  restricted \write18 enabled.
  file:line:error style messages enabled.
@@ -1727,24 +1727,24 @@ Underfull \hbox (badness 10000) in paragraph at lines 146--148
 
  []
 
-[6] [7] [8] [9] [10]
-Underfull \hbox (badness 10000) in paragraph at lines 270--272
+[6] [7] [8] [9] [10] [11]
+Underfull \hbox (badness 10000) in paragraph at lines 280--282
 
  []
 
-[11] [12]
+[12]
 Missing character: There is no ， in font [lmroman10-regular]:mapping=tex-text;!
- [13]
-Package atveryend Info: Empty hook `BeforeClearDocument' on input line 306.
-Package atveryend Info: Empty hook `AfterLastShipout' on input line 306.
+ [13] [14]
+Package atveryend Info: Empty hook `BeforeClearDocument' on input line 316.
+Package atveryend Info: Empty hook `AfterLastShipout' on input line 316.
  (./note.aux)
-Package atveryend Info: Empty hook `AtVeryEndDocument' on input line 306.
-Package atveryend Info: Empty hook `AtEndAfterFileList' on input line 306.
+Package atveryend Info: Empty hook `AtVeryEndDocument' on input line 316.
+Package atveryend Info: Empty hook `AtEndAfterFileList' on input line 316.
 
 
 LaTeX Font Warning: Some font shapes were not available, defaults substituted.
 
-Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 306.
+Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 316.
  ) 
 Here is how much of TeX's memory you used:
  38303 strings out of 493007
@@ -1755,4 +1755,4 @@ Here is how much of TeX's memory you used:
  1348 hyphenation exceptions out of 8191
  57i,8n,74p,10460b,492s stack positions out of 5000i,500n,10000p,200000b,80000s
 
-Output written on note.pdf (14 pages).
+Output written on note.pdf (15 pages).

note.tex

@@ -243,8 +243,18 @@ \subsubsection{非攻击性车}有$k$种颜色共$n$个车，第i种颜色有$n_
      \[0 \leqslant x_1 \leqslant n_1 , 0 \leqslant x_2 \leqslant n_2,\cdots , 0 \leqslant x_k \leqslant n_k\]
      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}

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}