3.1.1随机事件的概率的Breamer(2018-03-22)
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\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}