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公元1777年的一天，法国科学家布丰（D．Buffon1707-1788）的家里宾客满堂，原来他们是应主人的邀请前来观看一次奇特试验的。
试验开始，但见年已古稀的布丰先生兴致勃勃地拿出一张纸来，纸上预先画好了一条条等距离的平行线。
接着他又抓出一大把原先准备好的小针，这些小针的长度都是平行线间距离的一半。
然后布丰先生宣布：“请诸位把这些小针一根一根往纸上扔吧！不过，请大家务必把扔下的针是否与纸上的平行线相交告诉我。
”众宾哗然，一时议论纷纷，个个感到莫名其妙。
“圆周率π？这可是与圆半点也不沾边的呀！”布丰先生似乎猜透了大家的心思，得意洋洋地解释道：“诸位，这里用的是概率的原理，如果大家有耐心的话，再增加投针的次数，还能得到π的更精确的近似值。
不过，要想弄清其间的道理，只好请大家去看敝人的新作了。
”说着布丰先生扬了扬自己手上的一本《或然算术试验》的书。
π在这种纷纭杂乱的场合出现，实在是出乎人们的意料，然而它却是千真万确的事实。
由于投针试验的问题，是布丰先生最先提出的，所以数学史上就称它为布丰问题。
布丰得出的一般结果是：如果纸上两平行线间相距为d，小针长为，投针的次数为n，所投的针当中与平行线相交的次数是m，那么当n相当大时有：在上面故事中，针长等于平行线距离d的一半，所以代入上面公式简化我想，喜欢思考的读者，一定想知道布丰先生投针试验的原理，下面就是一个简单而巧妙的证明。
找一根铁丝弯成一个圆圈，使其直径恰好等于平行线间的距离d。
可以想象，对于这样的圆圈来说，不管怎么扔下，都将和平行线有两个交点。
因此，如果圆圈扔下的次数为n次，那么相交的交点总数必为2n。
现在设想把圆圈拉直，变成一条长为πd的铁丝。
显然，这样的铁丝扔下时与平行线相交的情形要比圆圈复杂些，可能有4个交点、3个交点、2个交点、1个交点，甚至于都不相交。
由于圆圈和直线的长度同为πd，根据机会均等的原理，当它们投掷次数较多，且相等时，两者与平行线组交点的总数可望是一样的。
课题：第六章 频率与概率2．投针试验课型：新授授课人：枣庄十五中学东校 王雪峰授课时间：2012年11月22日星期四教学目标：1. 能用试验的方法估计一些复杂的随机事件发生的概率．2. 经历试验、统计等活动过程，在活动中进一步发展学生的合作交流的意识和能力．3. 培养学生实事求是的科学态度．亲历试验，提高学生学习数学的兴趣． 教学重点：能用试验的方法估计一些复杂的随机事件发生的概率．教学难点：借助大量重复试验去感悟试验频率稳定于理论概率．教法及学法指导：教法：“试验—探究—归纳”法学法：学生通过小组活动，亲自动手试验，主动探索，发现规律；互动合作、解决问题；使学生主体地位得以体现．让学生充分感悟多次试验频率稳定于理论概率这一事实． 教具准备：大头针，图钉，多媒体演示教学过程:一．感悟导入[师]上节课我们介绍了用树状图或列表格的方法计算随机事件的概率．但这些方法也不是万能的。
下面我们来看一个例子．比如掷一枚图钉，有几种结果?它们是等可能的吗?[生]有“朝天”和“倾斜”两个可能结果，但我觉得这两个可能的结果不是等可能的.[师]能不能说“朝天”的概率是21，“倾斜”的概率也是21呢? [生]当然不能．[师]再例如，掷一只瓶盖，也有“正”“反”两种可能，但出现的可能性相等吗?[生]不相等．[师]很好．这些随机事件的概率还能用树状图或列表格的方法计算吗？该怎样求？[生]只有动手做大量的试验．因为我们知道：当实验次数很大时，实验频率稳定于理论概率，并可据此估计某一事件发生的概率．[师]看来，有些事件发生的概率只有亲自做很多次实验了．二．合作探究试验一：从一定高度落下的图钉，落地后可能钉尖着地，也可能钉帽着地．你估计哪种事件发生的概率大?试验目的：利用“当实验次数较大时，实验频率稳定于理论概率”来估计某一事件发生的概率．试验方式：小组合作交流，全班汇总实验数据，交流研讨．试验工具：形状、大小完全相同的图钉．试验步骤：1．分组：每组6人．2．每组每人做10次实验，根据实验结果，3．根据上表你认为哪种情况的频率较大?4．分别汇总本小组其中两人、三人、四人、五人、六人的实验数据，相应得到实验40次、60次、80次、100次时钉帽着地的频率，填写下表，并绘制折线统计图．5．汇总全班各小组其一个组．两个组、三个组、四个组……的实验数据，相应得到实验120次、180次、240次、360次……时钉帽着地的频率.6．估计钉帽着地的概率．注意：①图钉必须从一定高度自由落下，保证着地时的随机性；②组内同学合作时要进行适当的分工；③体现学生的自主性，实验活动以及实验数据的汇总等都可以由学生白行组织完成；④教师认真评价学生合作交流的意识和能力，学生的思维水平，学生的动手能力等.[师生共析]我们一同来研究一下，掷一枚图钉时，出现“钉帽着地”这一结果的概率．将图钉掷200次，每掷20次，统计一下两个组同学“钉帽着地”这一结果出现的次数，并算出相应的频率，如下表．将统计数据(“钉帽着地”的频率)从图中可发现，“顶帽着地”的频率开始“摆动”得很厉害，随着试验次数的增加，这个频率就开始比较稳定了，最后大致在56.5％左右摆动．由此我们可以估计“顶帽着地”的概率约为56．5％，即0.565.[师]在数学的历史上，有一个较为著名的投针实验：平面上画着一些平行线，相邻的两条平行线之间的距离为a，向此平面任投一长度为l(l<a)的针，该针可能与其中某一条平行线相交，也可能与它们都不相交．相交和不相交的可能性相同吗?你能通过列表或画树状图求出该针与平行线相交的概率吗?[生]相交和不相交的可能性不相同，由于结果的可能性不同，因此这个事件的概率也不能列表或画树状图求出该针与平行线相交的概率．也必须用“当实验次数较大时，实验频率稳定于理论概率”来估计该针与平行线相交的概率．[师]很好，我们还是分组活动．试验二：平面上画着一些平行线，相邻的两条平行线之间的距离都是a，向此平面任投一长度为l(l<a)的针，该针可能与其中某一条平行线相交，也可能与它们不相交，估计针与平行线相交的概率．试验目的：利用“当实验次数较大时，实验频率稳定于理论概率”，并据此估计针与平行线相交的概率．试验方式：小组交流，全班研讨的方法．试验工具：每组学生要在平面上画有相同距离“的一组平行线，并且有长度都为l的针(l<a)．要求针必须粗细均匀．活动步骤：1．分组，两人一组．2．取一张白纸，在上面画一组平行线．它们之间的距离为2厘米，另外准备一根1厘米长的针．在纸下面垫一层软布，使针落在纸面上时不会弹跳起来．3．每组至少完成50次实验，分别记录下其中相交和不相交的次数．4．统计全班的实验数据，估计针与平行线相交的概率．活动效果：在具体实验的过程中，要求每组学生都确定相同的l和a，而对于针可由教师统一准备．这样做是因为如果l和a取不同的值，实验结果是不同的．那样全班就无法统计数据．为了保证随机性。
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132.1129.9146.841.939101.140.230.757997.4 129.8127.5149.842.629.510140.840.259095.7 128.2126.3151.442.72510144.243.257697.3 129.7127.6152.543.122.2101.146.944.357197.8 129.4127.6152.143.319.7101.249.333.557097.9 129127.215343.618.1101.351.138.356798.1 128.1126153.642.116.7101.452.63856698.2 127.5125.7153.743.115.6101.453.837.656498.4 128.8127154.343.614.6101.554.938.356298.4 122119.4153.542.414.2101.555.437.960599.5 123.2120.115342.514.1101.655.63860499.5 122119.3151.942.814.1101.655.738.160599.6 122.3119.8151.14313.9101.655.737.960299.7 123.3120.6150.243.113.6101.65637.960099.8 122119.7152.943.713.6101.656.43860299.8 123120.1151.842.613.7101.656.437.960099.8 127.2124.3138.442.743.4101.433.836.662396.3 127.8124.7140.742.836.7101.436.537.761597.8 127.1124.2141.442.732.2101.538.937.761098.3 127.7124.9142.843.128.8101.640.937.561098.8 126.9124.1143.54326.3101.742.737.760898.8 162.5158.9166.143.323.3101.84538.3591100.9 162158.5166.843.921101.94737.6589101.1 161157.7167.243.318.110249.337.9589101.2160.6156.9167.443.316.21025138.1588101.2 159.8156.3168.443.215102.152.338.6588101.2 159.7156167.942.614.5102.152.938.5586101.5 159.1155.5168.642.313.7102.253.938.2584101.6 160.3156.6167.942.813.3102.254.537.5583101.6 159.3155.8169.743.713.1102.254.937.8583101.6 157.7154.216943.312.6102.355.638.3584101.4 157.7154.1167.941.912.2102.356.137.6583101.4 159155.8168.242.511.9102.356.637.9582101.6 158.9155.216943.511.7102.456.838.2581101.6 158.7155.1167.143.311.1102.357.337.6580101.7 159.2155.7166.743.510.7102.45837.7580101.5 141.513816043.611102.457.538.1592101.7 142.4138.7159.843.711.7102.456.837.8592101.7 142.7139.4158.643.111.7102.456.438.1590101.7 141.9138.51594311.7102.356.138590101.7 142.1138.6159.443.111.6102.355.537.7590101.7 141.8138.2160.443.111.6102.355.338.1589101.8 140.7137.4159.142.811.6102.355.137.7590101.7 142.5139.1159.343.111.4102.35538.1590101.7 140.2136.8159.142.711.2102.355.237.8589101.7 139.8136.4160.743.211.5102.354.838.1590101.7 141.4137.9159.143.112.1102.354.637.5587101.7 141.5138159.643.112.2102.354.637.6589101.7 167.1162.9175.143.411.9102.35537.5577103.6 1671631754311.7102.355.538.2575103.8 167.5162.9175.543.211.4102.455.738.4576103.9 166.7162.6177.243.811.3102.455.937.5575103.8 167.3163.2175.844.411.1102.456.237.8577103.6 166.3162.3175.644.911102.456.538.3578103.5 165.8161.8176.444.710.9102.456.738579103.3 168.4163.9174.644.110.6102.456.937.6580103.3 165.2161.3176.243.910.8102.456.938.4578103.4 166161.9175.144.510.7102.456.938580103.3 165.8161.3164.842.431.1101.632.939.3604102.2 166161.6167.142.826.4101.736.138.8592103.7 166.6162.316943.223101.83937.1589104.6 167.4162.9170.243.219.6101.941.537.4588104.9 166.3161.7170.843.117.310243.837.5586105.1 88.187.2114.442.51510246.138.154597.3 87.987.411542.613.2102.147.238.154996.7 87.286.5115.342.512.3102.24837.954896.8 87.487.1114.74311.6102.248.837.854696.9 87.687.1115.14311.7102.349.537.754197.5 86.185.5113.642.511.1102.350.437.653498.4 88.187.1113.943.711.2102.350.637.355495.5 87.487.1114.744.210.9102.350.937.254596.7 87.58711544.310.8102.351.337.454396.8 86.885.9115.943.110.6102.351.23854396.8 86.786.1116.14210.2102.351.338.254397 85.885.3116.341.79.9102.352.138.354296.8 114.2113.5135.143.89.4102.35437.645298.4 113.3113134.643.88.7102.455.737.4438100.5 113.1113.3133.444.38.4102.557.137.845298 113113.2134.642.58102.558.137.845397.9112.1112135.342.77.6102.558.837.945297.7 112.2111.9134.743.27.6102.559.337.745397.9 111.6111.3135.143.37.2102.659.83844898.4 111.4111.9135.1437.1102.659.838.344399.2 112.1112.3135.242.7 6.9102.66038.644698.7排气背压 k进气负压 k记录时间进气温度 ℃涡后温度 ℃进水温度 ℃出水温度 ℃84.786.819537.30.7-0.12010-09-16 15:54:4879.280.419838.10.6-0.12010-09-16 16:09:4880.985.820038.20.7-0.12010-09-16 16:24:4940.258.520634.30.7-0.12010-09-17 09:39:25648121336.40.6-0.12010-09-17 09:49:2635.48121237.60.7-0.12010-09-17 10:16:4777.483.621938.21-0.12010-09-17 14:17:3078.381.922339.40.9-0.12010-09-17 14:32:3084.281.322440.11-0.12010-09-17 14:47:3045.581.532240.3 1.1-0.12010-09-17 14:57:30748332841.1 1.4-0.12010-09-17 15:07:3172.883.233041.4 1.5-0.12010-09-17 15:17:3156.581.629442 1.9-0.12010-09-17 15:32:3181.683.729441.6 1.7-0.12010-09-17 15:47:3138.580.829441.3 1.7-0.12010-09-17 16:02:3176.384.137542.8 2.5-0.12010-09-17 16:12:3177.98536540.6 2.3-0.12010-09-18 09:49:5135.381.738250.7 2.4-0.22010-09-19 16:13:0332.281.433342.6 2.7-0.42010-09-20 09:42:1134.881.533643 2.9-0.42010-09-20 09:57:1236.480.933643.73-0.32010-09-20 10:12:1244.778.338346.24-0.52010-09-20 10:22:1248.682.438046.5 4.1-0.52010-09-20 10:32:1250.382.537846.3 4.2-0.52010-09-20 10:42:1249.282.536448.4 5.8-0.92010-09-20 10:52:1259.383.236248.9 5.8-0.92010-09-20 11:02:1357.282.936048.2 6.2-0.82010-09-20 11:12:135882.936048.8 5.9-0.92010-09-20 11:22:1359.583.234043.86-12010-09-21 07:55:1850.382.634945.4 5.7-0.92010-09-21 08:05:1949.282.840247.38.5-1.32010-09-21 08:15:1955.182.839052.48.6-1.12010-09-21 13:55:5242.882.340454.38.4-1.12010-09-21 14:05:5246.982.441154.78.6-1.12010-09-21 14:15:5243.282.341455.58.6-1.32010-09-21 14:25:5244.682.341655.28.5-1.12010-09-21 14:35:5345.680.541555.98.3-12010-09-21 14:45:5347.982.541554.58.6-1.12010-09-21 14:55:5348.182.541655.98.4-12010-09-21 15:05:5358.683.238257.79.3-1.32010-09-21 15:15:5352.482.838057.59.5-1.32010-09-21 15:25:5352.582.837854.89.7-1.42010-09-21 15:35:545382.237855.49.7-1.22010-09-21 15:45:5456.482.837753.29.3-1.42010-09-21 15:55:5453.382.837856.69.6-1.42010-09-21 16:05:5453.382.837555.59.6-1.42010-09-21 16:15:5433.282.335742.89.9-1.62010-09-22 09:15:3736.782.436444.39.8-1.62010-09-22 09:25:3839.182.537145.69.5-1.42010-09-22 09:35:384182.537445.89.8-1.52010-09-22 09:45:3842.682.637646.19.5-1.52010-09-22 09:55:3843.382.843555.113.7-1.82010-09-22 10:05:3845.482.943749.513.7-1.92010-09-22 10:15:3844.182.74369413.6-1.72010-09-22 10:25:3942.581.443696.313.7-1.72010-09-22 10:35:3942.882.843552.313.7-1.82010-09-22 10:45:3943.582.943466.313.8-1.82010-09-22 10:55:39 43.882.943454.913.7-1.82010-09-22 11:05:39 4482.84336713.8-1.72010-09-22 11:15:40 44.18243451.813.7-1.92010-09-22 11:25:40 44.282.943349.513.6-1.82010-09-22 11:35:40 43.982.943148.213.5-1.82010-09-22 11:45:4043.782.943148.913.9-1.62010-09-22 11:55:4044.182.743349.813.7-1.82010-09-22 12:05:40 4481.643148.413.8-1.72010-09-22 12:15:41 44.382.943051.213.6-1.82010-09-22 12:25:41 41.582.739950.612.8-1.82010-09-22 12:35:41 41.282.739849.913-1.82010-09-22 12:45:41 4182.639749.712.9-1.72010-09-22 12:55:41 40.782.139848.712.9-1.82010-09-22 13:05:41 40.978.939952.413-1.72010-09-22 13:15:42 40.882.740052.812.7-1.82010-09-22 13:25:42 40.682.639949.312.8-1.72010-09-22 13:35:42 40.382.639949.413.1-1.92010-09-22 13:45:42 40.581.839949.312.7-1.82010-09-22 13:55:42 40.578.140151.712.8-1.82010-09-22 14:05:42 40.482.740049.512.9-1.82010-09-22 14:15:43 40.382.640151.312.7-1.72010-09-22 14:25:4342.582.94495216.5-2.12010-09-22 14:35:4343.882.844951.516.4-1.92010-09-22 14:45:4344.381.145053.417.1-2.22010-09-22 14:55:43 46.382.945154.116.1-2.22010-09-22 15:05:44 45.182.945051.516.6-2.12010-09-22 15:15:44 458345050.816.1-22010-09-22 15:25:44 45.182.945154.316.5-2.22010-09-22 15:35:44 45.381.744952.116.9-2.12010-09-22 15:45:44 49.183.345152.516.1-2.22010-09-22 15:55:44 45.682.945048.616.5-2.12010-09-22 16:05:45 5983.443340.416.9-2.32010-09-23 09:23:11 74.785.444242.916.7-2.22010-09-23 09:33:11 76.58544644.316.5-2.32010-09-23 09:43:11 76.4854494516.2-2.22010-09-23 09:53:12 76.38545147.916.2-2.12010-09-23 10:03:12 68.182.837343.45-0.62010-09-23 10:13:1270.183.237242.6 4.6-0.72010-09-23 10:23:1271.383.437243.2 5.2-0.62010-09-23 10:33:1272.883.537243.2 5.1-0.62010-09-23 10:43:13 77.183.937243.8 5.2-0.62010-09-23 10:53:13 8283.637043 5.2-0.62010-09-23 11:03:13 76.384.437143.6 4.9-0.72010-09-23 11:13:13 73.68437145.2 5.1-0.72010-09-23 11:23:13 73.483.937145.3 5.1-0.72010-09-23 11:33:1473.183.937044.45-0.62010-09-23 11:43:1474.284.237046.8 5.2-0.62010-09-23 11:53:14 72.983.536946.2 4.9-0.62010-09-23 12:03:14 78.984.445346.4 5.3-0.72010-09-23 12:13:14 83.284.245545.2 5.4-0.72010-09-23 12:23:15 61.782.745244.7 5.6-0.62010-09-23 12:33:15 69.583.945045.4 5.5-0.72010-09-23 12:43:1568.483.445045.6 5.6-0.62010-09-23 12:53:1569.483.645045.4 5.6-0.72010-09-23 13:03:15 74.58444945.7 5.5-0.72010-09-23 13:13:15 80.783.744846.3 5.6-0.72010-09-23 13:23:16 82.387.344746.7 5.6-0.72010-09-23 13:33:16。
