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3 %\includeonlyframes{Representation}%Acknowledgement
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66 \usepackage{tabu}
67 \usepackage{graphics}
68 \usepackage{xcolor}
69 \usepackage[no-math]{fontspec}%--------------------------------------------------提供字体选择命令
70 \usepackage{xunicode}%-----------------------------------------------------------提供Unicode字符宏
71 \usepackage{xltxtra}%------------------------------------------------------------提供了针对XeTeX的改进并且加入了XeTeX的LOGO
72 \usepackage[BoldFont,SlantFont,CJKchecksingle]{xeCJK}%---------------------------使用xeCJK宏包
73 %================================== 设置中文字体 ================================%
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86 %================================== 设置英文字体 ================================%
87
88 %================================== 设置数学字体 ================================%
89 %\setmathsfont(Digits,Latin,Greek)[Numbers={Lining,Proportional}]{Minion Pro}
90 %================================== 设置数学字体 ================================%
91 \punctstyle{kaiming}%------------------------------------------------------------开明式标点格式
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93 \usepackage{tikz}
94 \usetikzlibrary{positioning,backgrounds}
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100 \pgfdeclarelayer{foreground}
101 \pgfsetlayers{background,main,foreground}
102 \usepackage{xifthen}
103 \usepackage{colortbl,dcolumn}
104 \usepackage{enumerate}
105 \usepackage{pifont}
106 \usepackage{tabularx}
107 \usepackage{booktabs}
108
109 %=================================== 数学符号 =================================%
110 \newcommand{\rtn}{\mathrm{\mathbf{R}}}
111 \newcommand{\N}{\mathrm{\mathbf{N}}}
112 \newcommand{\As}{\mathrm{a.s.}}
113 \newcommand{\Ae}{\mathrm{a.e.}}
114 \newcommand*{\PR}{\mathrm{\mathbf{P}}}
115 \newcommand*{\EX}{\mathrm{\mathbf{E}}}
116 \newcommand{\EXlr}[1]{\mathrm{\mathbf{E}}\left[#1\right]}
117 \newcommand*{\dif}{\,\mathrm{d}}
118 \newcommand*{\F}{\mathcal{F}}
119 \newcommand*{\h}{\mathcal{H}}
120 \newcommand*{\vp}{\varepsilon}
121 \newcommand*{\prs}{\dif\PR-\As}
122 \newcommand*{\dte}{\dif t-\Ae}
123 \newcommand*{\pts}{\dif\PR\times\dif t-\Ae}
124 \newcommand{\Ito}{It\^{o}}
125 \newcommand{\tT}[1][0]{[#1,T]}
126 \newcommand{\intT}[2][T]{\int^{#1}_{#2}}
127 \newcommand{\intTe}[1][t]{\intT[t+\varepsilon]{#1}}
128 \newcommand{\s}{\mathcal{S}}
129 \newcommand{\me}{\mathrm{e}}
130 \newcommand{\one}[1]{{\bf 1}_{#1}}
131 \renewcommand{\M}{{\rm M}}
132 \newcommand{\Me}[1][t]{M^{\varepsilon}_{#1}}
133 \newcommand{\Ne}[1][t]{N^{\varepsilon}_{#1}}
134 \newcommand{\Pe}[1][t]{P^{\varepsilon}_{#1}}
135 \DeclareMathOperator*{\sgn}{sgn}
136 %=================================== 数学符号 =================================%
137
138 \graphicspath{{figures/}}
139
140 \title[随机事件的概率]{\Huge\bfseries 3.1.1 随机事件的概率}
141
142 \author[wsx]{wsx}
143
144 \institute[lpssz]{\includegraphics[width=2cm]{lpssz}\\ lpssz}
145
146 \date{2018-03-22}
147
148 \setlength{\baselineskip}{22pt}
149 \renewcommand{\baselinestretch}{1.4}
150
151 \begin{document}
152
153 \setlength{\abovedisplayskip}{1ex}%------------------------------------------ 公式前的距离
154 \setlength{\belowdisplayskip}{1ex}%------------------------------------------ 公式后的距离
155
156 %\includeonlyframes{Brown}
157
158 \begin{frame}
159 \titlepage
160 \end{frame}
161 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
162 \begin{frame}{\Huge 第三章\quad 概率}
163 \begin{figure}
164 \vskip-6.5em% \hskip-6em
165 \includegraphics[page=1,scale=0.60]{mindmap}
166 \end{figure}
167 \end{frame}
168 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
169 \begin{frame}{\Huge 概率论的由来}
170 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.
171 \begin{figure}
172 \includegraphics[scale=0.4]{paris}
173 \end{figure}
174 \end{frame}
175 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
176 \begin{frame}{\Huge 概率论的由来}
177 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
178 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？
179 或者采用分摊，怎么分摊才合理呢？
180 \end{frame}
181 \begin{frame}{\Huge 3.1.1 随机事件的概率}
182 \begin{figure}
183 \vskip-6.5em% \hskip-6em
184 \includegraphics[page=2,scale=0.60]{mindmap}
185 \end{figure}
186 \end{frame}
187 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
188 \begin{frame}{\Huge 情景引入}
189 有四张红色卡片，分别编号为1,2,3,4，从中抽取一张卡片
190 \begin{itemize}
191 \item “抽到的卡片是红色的”
192 \item “抽到的卡片是蓝色的”
193 \item “抽到的卡片编号是1号”
194 \end{itemize}
195 \end{frame}
196 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
197 \begin{frame}{\Huge 事件的分类}
198 \begin{itemize}\pause
199 \item \alert{必然事件}：在条件S下，一定会发生的事件.\pause
200 \item \alert{不可能事件}：在条件S下，一定不会发生的事件.\pause
201 \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
202 \item {\centering 必然事件和不可能事件称为\alert{确定事件}.}\pause
203 \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
204 \end{itemize}
205
206 \end{frame}
207 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
208 \begin{frame}{\Huge 事件的分类 }
209 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？\\
210 %\begin{enumerate}
211 (1)“抛一块石头，下落”；\\
212 (2)“某人射击一次,中靶”;\\
213 (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”;\\
214 (4)“掷一枚硬币,出现正面”;\\
215 (5)“没有水分,种子能发芽”;\\
216 (6)“在常温下,焊锡熔化”
217 %\end{enumerate}
218 \end{frame}
219 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
220 \begin{frame}{\Huge 随机事件的概率}
221 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
222 \begin{block}{}
223 用\alert{概率}度量随机事件发生的可能性大小.
224 \end{block}\pause
225 如何获得随机事件发生的概率？
226 \end{frame}
227 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
228 \begin{frame}{\Huge 试验}
229 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：\\
230 \begin{center}
231 \setlength{\extrarowheight}{1.5mm}
232 %\addtolength{\tabcolsep}{1mm}
233 \rowcolors[]{1}{orange!70}{white!90!gray}
234 \begin{tabular}{|c|c|c|c|}
235 姓名 & 试验次数&正面朝上的次数&正面朝上的比例 \\\hline
236 & & & \\\hline
237 \end{tabular}
238 \end{center}
239 {\Huge \bfseries 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？
240
241 \end{frame}
242 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
243 \begin{frame}{\Huge 试验}
244 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：\\
245 \begin{center}
246 \setlength{\extrarowheight}{1.5mm}
247 %\addtolength{\tabcolsep}{1mm}
248 \rowcolors[]{1}{orange!70}{white!90!gray}
249 \begin{tabular}{|c|c|c|c|}
250 组次 & 试验总次数&正面朝上的总次数&正面朝上的比例 \\\hline
251 & & & \\\hline
252 \end{tabular}
253 \end{center}
254 {\Huge \bfseries 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？
255 \end{frame}
256 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
257 \begin{frame}{\Huge 试验}
258 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：\\
259 \begin{center}
260 \setlength{\extrarowheight}{1.5mm}
261 %\addtolength{\tabcolsep}{1mm}
262 \rowcolors[]{1}{orange!70}{white!90!gray}
263 \begin{tabular}{|c|c|c|c|}
264 班级 & 试验总次数&正面朝上的总次数&正面朝上的比例 \\\hline
265 & & & \\\hline
266 \end{tabular}
267 \end{center}
268 \end{frame}
269 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
270 \begin{frame}{\Huge 试验}
271 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.\\
272 \vskip6.5em
273 {\Huge \bfseries 观察 } \quad 这个条形图有什么特点？
274
275 \end{frame}
276 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
277 \begin{frame}{\Huge 试验}
278 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.\\
279 \vskip6.5em
280 {\Huge \bfseries 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？
281 \end{frame}
282 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
283 \begin{frame}{\Huge 基本概念}
284 \begin{block}{频数}
285 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.
286 \end{block}\pause
287 \begin{block}{频率}
288 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.
289 \end{block}
290 \end{frame}
291
292 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
293 \begin{frame}{\Huge 计算机模拟掷硬币试验}
294 \begin{figure}
295 \includegraphics[scale=0.25]{computer}
296 \end{figure}\pause
297 \begin{block}{}
298 在大量重复试验后,随着试验次数的增加，“正面朝上”的频率
299 \alert{逐渐稳定}在0.5的附近.
300 \end{block}
301 \end{frame}
302
303 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
304 \begin{frame}{\Huge 历史上一些掷硬币的试验结果}
305 \begin{center}
306 \setlength{\extrarowheight}{1.5mm}
307 %\addtolength{\tabcolsep}{1mm}
308 \rowcolors[]{1}{orange!70}{white!90!gray}
309 \begin{tabular}{|c|c|c|}
310 试验次数&正面朝上的次数&正面朝上的比例 \\\hline
311 2048 &1061 & 0.5181 \\
312 4040&2048&0.5069\\
313 12000&6019&0.5016\\
314 24000&12012&0.5005\\
315 30000&14984&0.4995\\
316 72088&36124&0.5011\\\hline
317 \end{tabular}
318 \end{center}
319 \end{frame}
320 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
321 \begin{frame}{\Huge 规律总结 }
322 由以上试验的规律，得到在一般情况下随机事件A的规律：\\
323 \begin{block}{}
324 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$P(A)$.\\
325 \end{block}
326 \begin{block}{大数定律}
327 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.}.
328 \end{block}
329 \end{frame}
330 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
331 \begin{frame}{\Huge 例题 }
332 例1 某运动员在同一条件下进行射击,结果如下表所示:
333 \begin{center}
334 \setlength{\extrarowheight}{1.5mm}
335 %\addtolength{\tabcolsep}{1mm}
336 \rowcolors[]{1}{orange!70}{white!90!gray}
337 \begin{tabular}{|c|c|c|c|c|c|c|}
338 射击次数$n$ &10 &20& 50& 100& 200& 500\\\hline
339 击中靶心次数$n_A$& 8& 19& 44& 92& 178& 455\\\hline
340 击中靶心的频率&&&&&& \\\hline
341 \end{tabular}
342 \end{center}
343
344
345 \begin{block}{}
346 (1)填写表中击中靶心的频率;
347 \end{block}
348 \begin{block}{}
349 (2)这个运动员射击一次,击中靶心的概率约是多少?
350 \end{block}
351 \end{frame}
352
353
354
355
356
357 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
358 \begin{frame}{\Huge 小组讨论}
359 %\Lauge
360
361 \begin{itemize}
362 \item 事件$A$发生的\alert{频率}是不是不变的？
363 \item 事件$A$发生的\alert{概率}是不是不变的？
364 \item 它们之间有什么\alert{区别与联系}？
365 \end{itemize}
366 \end{frame}
367 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
368 \begin{frame}{\Huge 区别与联系}\pause
369
370 \begin{block}{区别：}
371 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.
372 \end{block}\pause
373
374 \begin{block}{联系：}
375 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.
376 \end{block}
377 \end{frame}
378 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
379
380
381
382 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
383 \begin{frame}{\Huge 作业}
384 \Huge 课本P113练习1，2
385 \end{frame}
386
387
388 \begin{frame}{}
389 \begin{center}
390 \begin{tikzpicture}
391 \node[above,xscale=1.2,yscale=1.4]{\Huge\bfseries 谢谢!};
392 \node[xscale=1.2,above,yscale=-1.4,scope fading=south,opacity=0.2]{\Huge\bfseries 谢谢!};
393 \end{tikzpicture}
394 \end{center}
395 \end{frame}
396
397 \end{document}
