3.1.1随机事件的概率的Breamer(2018-03-22)

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	itle[随机事件的概率]{Hugefseries 3.1.1 随机事件的概率}

author[wsx]{wsx}

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date{2018-03-22}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
egin{frame}{Huge 第三章quad 概率}
egin{figure}
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includegraphics[page=1,scale=0.60]{mindmap}
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egin{frame}{Huge 概率论的由来}
相传，两位著名的数学家Pascal和Fermat经常在巴黎的咖啡屋里碰面讨论深奥的数学问题，为了解乏，他们经常玩一个简单的游戏.
egin{figure}
includegraphics[scale=0.4]{paris}
end{figure}
end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 概率论的由来}
重复抛掷硬币---每次正面朝上Pascal得一分，反面朝上Fermat得一分.一方先得三分，另一方就买单.\pause
一天，他们抛掷一次硬币，出现正面朝上（Pascal得一分）, Fermat有急事必须离开. 之后他们就想应该由谁来买单呢？ 
 或者采用分摊，怎么分摊才合理呢？
end{frame}
egin{frame}{Huge 3.1.1 随机事件的概率}
egin{figure}
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includegraphics[page=2,scale=0.60]{mindmap}
end{figure}
end{frame}
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egin{frame}{Huge 情景引入}
有四张红色卡片，分别编号为1,2,3,4，从中抽取一张卡片
 egin{itemize}
item “抽到的卡片是红色的”
 item “抽到的卡片是蓝色的” 
item “抽到的卡片编号是1号”
  end{itemize}
end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 事件的分类}
 egin{itemize}pause
item alert{必然事件}：在条件S下，一定会发生的事件.pause
    item alert{不可能事件}：在条件S下，一定不会发生的事件.pause
    item alert{随机事件}：在条件S下，可能发生也可能不发生的事件.pause
item {centering 必然事件和不可能事件称为alert{确定事件}.}pause
item 确定事件和随机事件统称为alert{事件},一般用大写字母$A,B,C$…表示.pause
  end{itemize}

end{frame}
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egin{frame}{Huge 事件的分类 }
 判断下列事件哪些是必然事件，哪些是不可能事件，哪些是随机事件？\
%egin{enumerate}
(1)“抛一块石头，下落”；\
(2)“某人射击一次,中靶”;\
(3)“从分别标有1,2,3,4,5的5张标签中任取1张,得到4号签”;\
(4)“掷一枚硬币,出现正面”;\
(5)“没有水分,种子能发芽”;\
(6)“在常温下,焊锡熔化”
%end{enumerate}
end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 随机事件的概率}
对于随机事件，我们知道它发生的可能性的大小事非常重要的。那么我们用什么来度量随机事件发生的可能性的大小呢？\ pause
egin{block}{}
用alert{概率}度量随机事件发生的可能性大小.
  end{block}pause
如何获得随机事件发生的概率？
end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 试验}
第一步，全班同学各取一枚相同的硬币，做$10$次抛硬币试验，每人记录下试验结果，填在下表中：\
egin{center}
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  %addtolength{	abcolsep}{1mm}
  
owcolors[]{1}{orange!70}{white!90!gray}
   egin{tabular}{|c|c|c|c|}
     姓名         & 试验次数&正面朝上的次数&正面朝上的比例 \hline
           & &  & \hline
    end{tabular}
 end{center}
{Huge fseries 思考 } quad 与其他同学的试验结果比较，你的结果和他们一致吗？为什么会出现这样的情况？

end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 试验}
第二步，每个小组把本组同学的试验结果统计一下，填入下表：\
egin{center}
  setlength{extrarowheight}{1.5mm}
  %addtolength{	abcolsep}{1mm}
  
owcolors[]{1}{orange!70}{white!90!gray}
   egin{tabular}{|c|c|c|c|}
     组次        & 试验总次数&正面朝上的总次数&正面朝上的比例 \hline
           & &  & \hline
    end{tabular}
 end{center}
{Huge fseries 思考 } quad 与其他小组的试验结果比较，各种的结果一致吗？为什么？
end{frame}
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egin{frame}{Huge 试验}
第三步，请一个同学把全班同学的试验结果统计一下，填入下表：\
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  %addtolength{	abcolsep}{1mm}
  
owcolors[]{1}{orange!70}{white!90!gray}
   egin{tabular}{|c|c|c|c|}
     班级        & 试验总次数&正面朝上的总次数&正面朝上的比例 \hline
           & &  & \hline
    end{tabular}
 end{center}
end{frame}
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egin{frame}{Huge 试验}
第四步，请把全班每个同学的试验中正面朝上的次数收集起来，并用条形图表示.\
vskip6.5em
{Huge fseries 观察 } quad 这个条形图有什么特点？

end{frame}
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egin{frame}{Huge 试验}
第五步，请同学们找出抛硬币时“正面朝上”这个事件发生的规律性.\
vskip6.5em
{Huge fseries 探究 } quad 如果同学们再重复一次上面的试验，全班的汇总结果还会和这次的汇总结果一致吗？如果不一致，你能说出原因吗？
end{frame}
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egin{frame}{Huge 基本概念}
egin{block}{频数}
在相同的条件$S$下重复$n$次试验，观察某一个事件$A$是否出现，称$n$次试验中事件A出现的次数$n_A$为事件$A$出现的频数.
  end{block}pause
egin{block}{频率}
我们称事件$A$出现的比例$f_n(A)=dfrac{n_A}{n}$为事件出现的频率.
  end{block}
end{frame}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 计算机模拟掷硬币试验}
egin{figure}
includegraphics[scale=0.25]{computer}
end{figure}pause
egin{block}{}
在大量重复试验后,随着试验次数的增加，“正面朝上”的频率
alert{逐渐稳定}在0.5的附近.
  end{block}
end{frame}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 历史上一些掷硬币的试验结果}
egin{center}
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owcolors[]{1}{orange!70}{white!90!gray}
   egin{tabular}{|c|c|c|}
      试验次数&正面朝上的次数&正面朝上的比例 \hline
       2048    &1061 & 0.5181 \
4040&2048&0.5069\
12000&6019&0.5016\
24000&12012&0.5005\
30000&14984&0.4995\
72088&36124&0.5011\hline
    end{tabular}
 end{center}
end{frame}
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egin{frame}{Huge 规律总结 }
由以上试验的规律，得到在一般情况下随机事件A的规律：\
 egin{block}{}
一般来说,在大量重复进行同一试验时,随着试验次数的增加,事件A发生的频率$f_n(A)=dfrac{n_A}{n}$会逐渐稳定在区间$[0,1]$中的某个alert{常数}上.pause 这个常数叫做事件A的alert{概率}，记作$P(A)$.\
  end{block}
  egin{block}{大数定律}
1713年，瑞士数学家雅各布·贝努利(Jacob Bernouli)对这一客观规律性从理论上给予了证明，并提出了著名的大数定律：alert{随着试验次数的增加，频率稳定在概率附近.}.
  end{block}
end{frame}
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egin{frame}{Huge 例题 }
例1 某运动员在同一条件下进行射击,结果如下表所示:
egin{center}
  setlength{extrarowheight}{1.5mm}
  %addtolength{	abcolsep}{1mm}
  
owcolors[]{1}{orange!70}{white!90!gray}
   egin{tabular}{|c|c|c|c|c|c|c|}
射击次数$n$    &10    &20&    50&    100&    200&    500\hline
击中靶心次数$n_A$&    8&    19&    44&    92&    178&    455\hline
击中靶心的频率&&&&&&    \hline
    end{tabular}
 end{center}

                    
 egin{block}{}
(1)填写表中击中靶心的频率;
  end{block}
  egin{block}{}
(2)这个运动员射击一次,击中靶心的概率约是多少?
  end{block}
end{frame}





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egin{frame}{Huge 小组讨论}
%Lauge

 egin{itemize}
    item 事件$A$发生的alert{频率}是不是不变的？
    item 事件$A$发生的alert{概率}是不是不变的？
item 它们之间有什么alert{区别与联系}？
  end{itemize}
end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
egin{frame}{Huge 区别与联系}pause

  egin{block}{区别：}
频率本身是随机的,在试验前不能确定,做同样次数或不同次数的重复试验得到的事件的频率都可能不同.而概率是一个确定数,是客观存在的,与每次试验无关.
  end{block}pause

  egin{block}{联系：}
随着试验次数的增加, 频率会在概率的附近摆动,并趋于稳定.在实际问题中,若事件的概率未知,常用频率作为它的估计值.
  end{block}
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egin{frame}{Huge 作业}
Huge 课本P113练习1，2
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