(共13张PPT)
人教版数学必修三
3.3.2均匀随机数的产生
复习回顾
计算随机事件发生概率的几种方法：
（2）用古典概型的计算公式来计算随机事件发生的概率。
（1）通过做试验或用计算机模拟试验等方法得到事件发生的频率，以此来估计概率；
（3）用几何概型的计算公式来计算随机事件发生的概率。
一、随机数的产生方法
1、实例法
如掷骰子、掷硬币、抽签、从一叠纸牌中抽牌、正多边形旋转器或钟表式图形转盘等等．
例如：掷硬币，1表示正面，0表示反面，连掷四枚硬币就可得到二进制数0000到1111，即十进制0～15.
2、计算器或计算机模拟法
(1)现在的大部分科学计算器都能产生0～1之间的均匀随机数(实数)，例如：
②具体操作步骤如下：
①利用计算器的RAND函数可以产生[0,1]上的均匀随机数，试验结果是区间[0,1]内的任意一个实数，而且出现任何一个实数是等可能的．
SHIFT
RAN#
=
SHIFT
RAN#*（b-a)+a
=
这样产生的是[0,1]之间的所有实数。那么
如何产生[a,b]之间的所有实数？
(2)计算机软件法：几乎所有的高级编程语言都有随机函数，借助随机函数可以产生一定范围的随机数．
例如：用Excel软件中产生[0,1]上的均匀随机数的函数rand( )来模拟．
若要产生[a，b]上的均匀随机数，可使用变换rand( )*(b－a)＋a，试验的结果是产生a～b之间的任何一个实数，并且出现a～b之间任何一个实数都是等可能的．
若要产生[a，b]上的整数随机数可用取整函数randbetween(a,b) 或 int(rand( )*(b－a)＋a。
进入\新建 Microsoft Excel 工作表.xls演示。
（一）用均匀随机模拟法估计几何概型
二、产生均匀随机数的实际意义
【例2】假设您家订了一份报纸,送报人可能在早上6:30—7:30之间把报纸送到你家,你父亲离开家去工作的时间在早上7:00—8:00之间,问你父亲在离开家前能得到报纸(称为事件A)的概率是多少
(1)解法1：利用几何概型公式．
提示：以横坐标X表示报纸送到时间,以纵坐标Y表示父亲离家时间建立平面直角坐标系,父亲在离开家前能得到报纸的事件构成区域是：
(2)解法2：用随机模拟方法．
进入\新建 Microsoft Excel 工作表.xls演示。
【点评】
用均匀随机数模拟的关键是把实际问题中事件A及基本事件总体对应的区域转化为随机数的范围．
用计算机产生随机数，这种方法既可以产生大量的随机数，又可以自动统计试验的结果，同时可以在短时间内多次重复试验，可以对试验结果的随机性和规律性有更深刻的认识。
【例3】在正方形中随机撒一把豆子，用随机模拟的方法估计圆周率的值．
豆子落在圆内的概率=
圆的面积
正方形的面积
≈
落在圆中的豆子数
落在正方形中的豆子数
圆的面积
正方形的面积
≈
落在圆中的豆子数
落在正方形中的豆子数
随着试验次数的增大，结果的精度会越来越高.
（二）用均匀随机模拟法估算圆周率π
进入\新建 Microsoft Excel 工作表.xls演示
【点评】
可以看到随着试验次数的增多，大多数估计值越来越接近概率值，但试验次数多并不一定就比次数少的精度高，体现出试验估计值的“随机性”。
同时我们可以看到1000次试验得到π的估计值精确度并不高，同时我们考虑到实际试验的时间和精力问题，因此一般采用计算机随机模拟。
【例4】利用随机模拟方法计算图中阴影部分(y=1和y=x2所围成的部分)的面积.
根据几何概型计算概率的公式,概率等于面积之比．如果概率用频率近似,在不规则图形外套上一个规则图形，则不规则图形的面积近似等于规则图形的面积乘以频率．而频率可以通过模拟的方法得到，从而得到了不规则图形面积的近似值．
本题套上的规则图形面积为2，所以本题所求的不规则图形的面积等于频率乘以2．
进入\新建 Microsoft Excel 工作表.xls演示
（三）用均匀随机模拟法计算不规则图形的面积
【点评】
解决本题的关键是利用随机模拟法和几何概型的概率公式分别求得几何概率。然后通过解方程求得不规则图形(M)的面积。
求不规则图形的面积，其一般方法为将不规则图形放在一个规则图形的内部（一般内接）然后利用两图形的面积比等于概率，如果概率用频率来近似，则不规则图形的面积近似等于规则图形的面积乘以频率。
三、总结反思
1、我学会用计算器产生[a,b]上的均匀随机数了吗？
2、想一想，这三个例题分别说明了什么问题？
例题2告诉我们可以利用随机模拟的方法估计几何概型中随机事件的概率值。
例题3、4告诉我们可以利用随机模拟的方法估计几何图形的面积，而当面积容易算出时进而可以估计其他未知量，这里的频率由随机试验获得，概率由几何概型得到。
谢谢大家！3．3．2<<均匀随机数的产生>>教案
执教者：黄杰
一、教学目标：
1、知识与技能：
（1）了解均匀随机数的概念；
（2）掌握利用计算机（计算器）产生均匀随机数的方法；
（3）会利用均匀随机数解决具体的有关概率的问题．
2、过程与方法：
通过模拟试验，感知应用数字解决问题的方法，自觉养成动手、动脑的良好习惯。
3、情感态度与价值观：
本节课的主要特点是随机试验多，学习时养成勤学严谨的学习习惯。
二、重点与难点：
利用计算机或计算器产生均匀随机数并运用到概率的实际应用中．
三、学法与教学用具：
1、通过对本节知识的探究与学习，感知用图形解决概率问题的方法，掌握数学思想与逻辑推理的数学方法；
2、教学用具：投灯片，计算机及多媒体教学．
四、教学设想：
1、课前回顾：
（1）几何概率模型：如果每个事件发生的概率只与构成该事件区域的长度（面积或体积）成比例，则称这样的概率模型为几何概率模型；
（2）几何概型的概率公式：
P（A）=；
（3）几何概型的特点：1）试验中所有可能出现的结果（基本事件）有无限多个；2）每个基本事件出现的可能性相等．
2、例题分析：
【例2】假设您家订了一份报纸,送报人可能在早上6:30—7:30之间把报纸送到你家,你父亲离开家去工作的时间在早上7:00—8:00之间,问你父亲在离开家前能得到报纸(称为事件A)的概率是多少
【解】以横坐标X表示报纸送到时间,以纵坐标Y表示父亲离家时间建立平面直角坐标系,父亲在离开家前能得到报纸的事件构成区域是：
由于随机试验落在方形区域内任何一点是等可能的,所以符合几何概型的条件.根据题意,只要点落到阴影部分,就表示父亲在离开家前能得到报纸,即事件A发生,所以
用计算机产生随机数模拟试验
1.选定A1格，键入“=RAND（）”，按Enter键，则在此格中的数是随机产生的[0，1]之间的均匀随机数．
2.选定A1格，按Ctrl+C快捷键，选定A2～A50,B1～B50，按Ctrl+V快捷键，则在A2～A50,B1～B50的数均为[0，1]之间的均匀随机数．用A列的数加7表示父亲离开家的时间，B列的数加6.5表示报纸到达的时间．这样我们相当于做了50次随机试验．
3.如果A+7>B+6.5，即A－B>－0.5，则表示父亲在离开家前能得到报纸．
4.选定D1格，键入“=A1-B1”；再选定D1，按Ctrl+C，选定D2～D50，按Ctrl+V．
5.选定E1格，键入频数函数“=FREQUENCY（D1:D50,－0.5）”,按Enter键，此数是统计D列中，比－ 0.5小的数的个数，即父亲在离开家前不能得到报纸的频数．
6.选定F1格，键入“=1－E1/50”，按Enter键，此数是表示统计50次试验中，父亲在离开家前能得到报纸的频率．
【例3】在正方形中随机撒一把豆子，用随机模拟的方法估计圆周率的值．
【分析：】
用计算机产生随机数模拟试验
1.选定A1，键入“=(rand( ) －0.5) ＊2”，表示得到[-1，1]之间的随机数．
2.选定A1格，按Ctrl+C，选定A2～A1000,B1～B1000，按Ctrl+V,则在A2～A1000,B1～B1000的数均为[ -1，1]之间的均匀随机数．
3.选定D1，键入“=power(A1，2)+power(B1，2)”，再选定D1，按“ctrl+C”，选定D2～D1000，按“ctrl+V”，则D列表示A2+B2．
4.选定F1,键入“=IF(D1>1，1，0)”，再选定F1，按“ctrl+C”,选定F2～F1000，按“ctrl+V”．则如果D列中A2+B2>1，F列中的值为1，否则F列中的值为0．
5.选定H1，键入“=frequency(F1：F10，0.5) ”,表示F1~F10中小于0.5的个数，即前10次试验中落在圆内的豆子数；类似地，分别得到表示前20次、50次、100次、500次、1000次试验中落在圆内的豆子数.
6.选定I1，键入“=H1*4/10”，表示根据前10次试验得到圆周率的估计值，类似地，可以得到根据前20次、50次、100次、500次和1000次试验得到的圆周率的估计值
【例4】利用随机模拟方法计算图中阴影部分(y=1和y=x2所围成的部分)的面积.
【分析】：根据几何概型计算概率的公式,概率等于面积之比．如果概率用频率近似,在不规则图形外套上一个规则图形，则不规则图形的面积近似等于规则图形的面积乘以频率．而频率可以通过模拟的方法得到，从而得到了不规则图形面积的近似值．
本题套上的规则图形面积为2，所以本题所求的不规则图形的面积等于频率乘以2．
用计算机产生随机数模拟试验
1.选定A1，键入“=(RAND( ) －0.5) *2”，表示得到[-1，1]之间的随机数．选定A1格，按Ctrl+C，选定A2～A1000,按Ctrl+V,则在A2～A1000的数均为[ -1，1]之间的均匀随机数.
2.选定B1，键入“=RAND( )”，表示得到[0，1]之间的随机数．选定B1格，按Ctrl+C，选定B2～B1000,按Ctrl+V,则在B1～B1000的数均为[0，1]之间的均匀随机数．
3.选定D1，键入“=B1－power(A1，2)”，再选定D1，按“ctrl+C”,选定D2～D1000，按“ctrl+V”，则D列表示B－A2.
4.选定F1,键入“=IF(D1>0，0，1)”，再选定F1，按“ctrl+C”,选定F2～F1000，按“ctrl+V”．则如果D列中元素大于0，则F列中的值为0，否则F列中的值为1．
5.选定H1，键入“=FREQUENCY(F1：F10，0.5) ”,表示F1～F10中小于0.5的个数，即前10次试验中落在阴影部分的频数；类似地，分别得到表示前20次、50次、100次、500次、1000次试验中落在阴影部分的频数.
6.选定I1，键入“=H1*2/10”，表示根据前10次试验得到阴影部分面积的估计值，类似地，可以得到根据前20次、50次、100次、500次和1000次试验得到阴影部分面积的估计值．
3、课堂小结：
均匀随机数在日常生活中，有着广泛的应用，我们可以利用计算器或计算机来产生均匀随机数，从而来模拟随机试验，其具体方法是：建立一个概率模型，它与某些我们感兴趣的量（如概率值、常数 ）有关，然后设计适当的试验，并通过这个试验的结果来确定这些量．
4、课堂练习：
1．某班有45个，现要选出1人去检查其他班的卫生，若每个人被选到的机会均等，则恰好选中学生甲主机会有多大？
2．曲线y=-x2+1与x轴、y轴围成一个区域A，直线x=1、直线y=1、x轴围成一个正方形，向正方形中随机地撒一把芝麻，利用计算机来模拟这个试验，并统计出落在区域A内的芝麻数与落在正方形中的芝麻数。
5、评价标准：
1．提示：本题应用计算器产生随机数进行模拟试验，请按照下面的步骤独立完成。
（1）用1～45的45个数来替代45个人；
（2）用计算器产生1～45之间的随机数，并记录；
（3）整理数据并填入下表
试 验次 数 50 100 150 200 250 300 350 400 450 500 600 650 700 750 800 850 900 1000 1050
1出现的频数
1出现的频率
（4）利用稳定后1出现的频率估计恰好选中学生甲的机会。
2．解：如下表，由计算机产生两例0~1之间的随机数，它们分别表示随机点（x,y）的坐标。如果一个点（x,y）满足y≤-x2+1，就表示这个点落在区域A内，在下表中最后一列相应地就填上1，否则填0。
x y 计数
0.598895 0.940794 0
0.512284 0.118961 1
0.496841 0.784417 0
0.112796 0.690634 1
0.359600 0.371441 1
0.101260 0.650512 1
… … …
0.947386 0.902127 0
0.117618 0.305673 1
0.516465 0.222907 1
0.596393 0.969695 0
6、课后作业：
课后练习及练习册相关内容。（1）、在A1中输入“=rand( )”即可产生[0,1]上的均匀随机数
按Ctrl+C快捷键（等同右键点击复制），选定区域如 A2～A50；
（2）、在C1中输入“=RANDBETWEEN(0,1)”得到[0,1]之间的
随机整数。同样可以在指定区域产生一系列随机整数。
（3）、在B1中输入“=rand（ ）*（b-a）+a”可以得到[a,b]之
间的均匀随机数。例如区间为[5,25]。
（4）、在D1中输入“=RANDBETWEEN(a,b)”得到[a,b]之间
的随机整数。
RAND() RAND()*20+5 RANDBETWEEN(0,1)
1、选定A1格，键入“=RAND（）”，按Enter键，则在此格中的数是随机产生的[0，1]之间的均匀随机数． 2、选定A1格，按Ctrl+C快捷键，选定A2～A50,B1～B50，按Ctrl+V快捷键，则在A2～A50,B1～B50的数均为[0，1]之间的均匀随机数．用A列的数加7表示父亲离开家的时间，B列的数加6.5表示报纸到达的时间．这样我们相当于做了50次随机试验． 3、如果A+7>B+6.5，即A－B>－0.5，则表示父亲在离开家前能得到报纸． 4、选定D1格，键入“=A1-B1”；再选定D1，按Ctrl+C，选定D2~D50，按Ctrl+V． 5、选定E1格，键入频数函数“=FREQUENCY（D1:D50,－0.5）”,按Enter键，此数是统计D列中比－ 0.5小的数的个数，即父亲在离开家前不能得到报纸的频数． 6、选定F1格，键入“=1－E1/50”，按Enter键，此数是表示统计50次试验中，父亲在离开家前能得到报纸的频率．
1.选定A1，键入“=(rand( )－0.5)*2”，表示得到[-1，1]之间的随机数． 2.选定A1格，按Ctrl+C，选定A2～A1000,B1～B1000，按Ctrl+V,则在A2～A1000,B1～B1000的数均为[ -1，1]之间的均匀随机数． 3.选定D1，键入“=power(A1，2)+power(B1，2)”，再选定D1，按“ctrl+C”，选定D2～D1000，按“ctrl+V”，则D列表示A +B ． 4.选定F1,键入“=IF(D1>1，1，0)”，再选定F1，按“ctrl+C” 选定F2～F1000，按“ctrl+V”．则如果D列中A +B >1，F列中的值为1，否则F列中的值为0． 5.选定H1，键入“=frequency(F1：F10，0.5) ”,表示F1～F10中小于0.5的个数，即前10次试验中落在圆内的豆子数；类似地，分别得到表示前20次、50次、100次、500次、1000次试验中落在圆内的豆子数. 6.选定I1，键入“=H1＊4/10”，表示根据前10次试验得到圆周率的估计值，类似地，可以得到根据前20次、50次、100次、500次和1000次试验得到的圆周率的估计值．
(rand( )－0.5)*2 power(A1，2)+power(B1，2) IF(D1>1，1，0) frequency(F1：F10，0.5)
0.5778534608 0.739643337 0.4057287149 0 8 1.6
0.2732126975 0.7978585074 0.7232133293 0 16 1.6
-0.1445277287 0.7091454999 0.6882572355 0 38 1.52 1.选定A1，键入“=(RAND( ) －0.5)＊2”，表示得到[-1，1]之间的随机数．选定A1格，按Ctrl+C，选定A2～A1000,按Ctrl+V,则在A2～A1000的数均为[ -1，1]之间的均匀随机数. 2.选定B1，键入“=RAND( )”，表示得到[0，1]之间的随机数．选定B1格，按Ctrl+C，选定B2～B1000,按Ctrl+V,则在B1～B1000的数均为[0，1]之间的均匀随机数． 3.选定D1，键入“=B1－power(A1，2)”，再选定D1，按ctrl+C 选定D2～D1000，按“ctrl+V”，则D列表示B－A . 4.选定E1,键入“=IF(D1>0，0，1)”，再选定E1，按ctrl+C,选定E2～E1000，按“ctrl+V”．则如果D列中元素大于0，则E列中的值为0，否则E列中的值为1． 5.选定F1，键入“=FREQUENCY(E1：E10，0.5) ”,表示E1～E10中小于0.5的个数，即前10次试验中落在阴影部分的频数；类似地，分别得到表示前20次、50次、100次、500次、1000次试验中落在阴影部分的频数. 6.选定G1，键入“=F1＊2/10”，表示根据前10次试验得到阴影部分面积的估计值，类似地，可以得到根据前20次、50次、100次、500次和1000次试验得到阴影部分面积的估计值．
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