[从头学数学] 第176节 概率
剧情提要：
[机器小伟]在[project师阿伟]的陪同下进入了结丹中期的修炼，
这次要修炼的目标是[概率]。
每次结果都不一样的。
[机器小伟]在[project师阿伟]的陪同下进入了结丹中期的修炼，
这次要修炼的目标是[概率]。
正剧開始：
星历2016年04月14日 09:46:28, 银河系厄尔斯星球中华帝国江南行省。
[project师阿伟]正在和[机器小伟]一起研究[概率]。
小伟决定自己来掷硬币玩一玩：
<span style="font-size:18px;">def tmp():
result = [];
print('试验次数--正面朝上次数---正面朝上频率');
for i in range(5, 105, 5):
count = 0;
for j in range(i):
number = random.randint(1, 10000);
if number%2 == 0:
count+=1;
#result.append([i, count, count/i]);
print('[',i,',',count,',',count/i, ']');
if __name__ == '__main__':
tmp();
>>>
试验次数--正面朝上次数---正面朝上频率
[ 5 , 1 , 0.2 ]
[ 10 , 3 , 0.3 ]
[ 15 , 9 , 0.6 ]
[ 20 , 9 , 0.45 ]
[ 25 , 15 , 0.6 ]
[ 30 , 11 , 0.36666666666666664 ]
[ 35 , 20 , 0.5714285714285714 ]
[ 40 , 14 , 0.35 ]
[ 45 , 20 , 0.4444444444444444 ]
[ 50 , 24 , 0.48 ]
[ 55 , 27 , 0.4909090909090909 ]
[ 60 , 28 , 0.4666666666666667 ]
[ 65 , 40 , 0.6153846153846154 ]
[ 70 , 35 , 0.5 ]
[ 75 , 38 , 0.5066666666666667 ]
[ 80 , 40 , 0.5 ]
[ 85 , 38 , 0.4470588235294118 ]
[ 90 , 47 , 0.5222222222222223 ]
[ 95 , 49 , 0.5157894736842106 ]
[ 100 , 55 , 0.55 ]
>>> ================================ RESTART ================================
>>>
试验次数--正面朝上次数---正面朝上频率
[ 5 , 2 , 0.4 ]
[ 10 , 4 , 0.4 ]
[ 15 , 10 , 0.6666666666666666 ]
[ 20 , 12 , 0.6 ]
[ 25 , 11 , 0.44 ]
[ 30 , 9 , 0.3 ]
[ 35 , 19 , 0.5428571428571428 ]
[ 40 , 20 , 0.5 ]
[ 45 , 24 , 0.5333333333333333 ]
[ 50 , 24 , 0.48 ]
[ 55 , 32 , 0.5818181818181818 ]
[ 60 , 29 , 0.48333333333333334 ]
[ 65 , 35 , 0.5384615384615384 ]
[ 70 , 33 , 0.4714285714285714 ]
[ 75 , 32 , 0.4266666666666667 ]
[ 80 , 39 , 0.4875 ]
[ 85 , 45 , 0.5294117647058824 ]
[ 90 , 47 , 0.5222222222222223 ]
[ 95 , 42 , 0.4421052631578947 ]
[ 100 , 51 , 0.51 ]
>>>
试验次数--正面朝上次数---正面朝上频率
[ 5 , 4 , 0.8 ] ,
[ 10 , 4 , 0.4 ] ,
[ 15 , 10 , 0.6666666666666666 ] ,
[ 20 , 14 , 0.7 ] ,
[ 25 , 10 , 0.4 ] ,
[ 30 , 13 , 0.43333333333333335 ] ,
[ 35 , 17 , 0.4857142857142857 ] ,
[ 40 , 26 , 0.65 ] ,
[ 45 , 22 , 0.4888888888888889 ] ,
[ 50 , 24 , 0.48 ] ,
[ 55 , 29 , 0.5272727272727272 ] ,
[ 60 , 30 , 0.5 ] ,
[ 65 , 32 , 0.49230769230769234 ] ,
[ 70 , 33 , 0.4714285714285714 ] ,
[ 75 , 34 , 0.4533333333333333 ] ,
[ 80 , 46 , 0.575 ] ,
[ 85 , 38 , 0.4470588235294118 ] ,
[ 90 , 47 , 0.5222222222222223 ] ,
[ 95 , 46 , 0.4842105263157895 ] ,
[ 100 , 58 , 0.58 ] ,</span>每次结果都不一样的。
看一看这些数究竟是如何的：
<span style="font-size:18px;"> if (1) {
var config = new PlotConfiguration();
config.init();
config.setPreference();
var r = 20;
//config.setSector(1,1,1,1);
//config.graphPaper2D(0, 0, r);
//config.axis2D(0, 0,180);
var array = [
[ 5 , 4 , 0.8 ] ,
[ 10 , 4 , 0.4 ] ,
[ 15 , 10 , 0.6666666666666666 ] ,
[ 20 , 14 , 0.7 ] ,
[ 25 , 10 , 0.4 ] ,
[ 30 , 13 , 0.43333333333333335 ] ,
[ 35 , 17 , 0.4857142857142857 ] ,
[ 40 , 26 , 0.65 ] ,
[ 45 , 22 , 0.4888888888888889 ] ,
[ 50 , 24 , 0.48 ] ,
[ 55 , 29 , 0.5272727272727272 ] ,
[ 60 , 30 , 0.5 ] ,
[ 65 , 32 , 0.49230769230769234 ] ,
[ 70 , 33 , 0.4714285714285714 ] ,
[ 75 , 34 , 0.4533333333333333 ] ,
[ 80 , 46 , 0.575 ] ,
[ 85 , 38 , 0.4470588235294118 ] ,
[ 90 , 47 , 0.5222222222222223 ] ,
[ 95 , 46 , 0.4842105263157895 ] ,
[ 100 , 58 , 0.58 ]
];
var len = array.length;
var stat = new Statistic();
var data = [], text = [];
//横坐标
for (var i = 0; i <len; i++) {
data.push(array[i][2].toFixed(2));
text.push(array[i][0].toFixed(0));
}
stat.init(data, '试验次数', '正面朝上频率', 1);
stat.rateHistogram(text, 0, 0);
}</span>
画个折线图瞅瞅：
<span style="font-size:18px;"> if (1) {
var config = new PlotConfiguration();
config.init();
config.setPreference();
var r = 20;
//config.setSector(1,1,1,1);
//config.graphPaper2D(0, 0, r);
//config.axis2D(0, 0,180);
var array = [
[ 5 , 4 , 0.8 ] ,
[ 10 , 4 , 0.4 ] ,
[ 15 , 10 , 0.6666666666666666 ] ,
[ 20 , 14 , 0.7 ] ,
[ 25 , 10 , 0.4 ] ,
[ 30 , 13 , 0.43333333333333335 ] ,
[ 35 , 17 , 0.4857142857142857 ] ,
[ 40 , 26 , 0.65 ] ,
[ 45 , 22 , 0.4888888888888889 ] ,
[ 50 , 24 , 0.48 ] ,
[ 55 , 29 , 0.5272727272727272 ] ,
[ 60 , 30 , 0.5 ] ,
[ 65 , 32 , 0.49230769230769234 ] ,
[ 70 , 33 , 0.4714285714285714 ] ,
[ 75 , 34 , 0.4533333333333333 ] ,
[ 80 , 46 , 0.575 ] ,
[ 85 , 38 , 0.4470588235294118 ] ,
[ 90 , 47 , 0.5222222222222223 ] ,
[ 95 , 46 , 0.4842105263157895 ] ,
[ 100 , 58 , 0.58 ]
];
var len = array.length;
var stat = new Statistic();
var data = [], text = [];
//横坐标
for (var i = 0; i <len; i++) {
data.push(array[i][2]);
text.push(array[i][0].toFixed(0));
}
stat.init(data, '试验次数', '正面朝上频率', 1);
stat.lineGraph(text, 0, 0, 0, 1.0);
}
//折线图
this.lineGraph = function(lableArray, xOffset, yOffset, yMin, yMax) {
lableArray = lableArray ? lableArray : [];
var lables = lableArray.length;
xOffset = xOffset ? xOffset : 0;
yOffset = yOffset ? yOffset : 0;
var colorArray = ['red', 'orange', '#0088FF', 'green', 'cyan', 'blue', '#FF00FF',
'#FF8844', 'purple'];
var colors = colorArray.length;
var height = 380, width = 580;
plot.save()
.translate(xOffset+60, yOffset+50);
plot.setLineWidth(2)
.setTextAlign('right');
var max = yMax ? yMax : this.max();
var min = yMin ? yMin : 0;
//最大值的末位为0的近似数，比方最大值25。最合适的近似数为30
var dy = Math.round((max-min)/10*100)/100;
var size = this.size();
var perH = Math.round((height-100) / 10);
var part = 2 * size;
var perW = Math.round((width-100) / part);
//宽和高度边界
var wBound = part*perW, hBound = 10*perH;
plot.setLineWidth(5)
.strokeRect(0, 0, wBound, hBound);
this.axis2D(0, hBound, wBound+20, hBound+20, this.xLabel, this.yLabel);
plot.setLineWidth(2);
var count = 0;
for (var i = hBound; i >-1; i -= perH) {
plot.fillText((min+dy*count).toFixed(2), -10, i+10, 30);
count++;
if (i > 0) {
plot.beginPath()
.moveTo(0, i)
.lineTo(wBound, i)
.closePath()
.stroke();
}
}
for (var i = 0; i < part; i++) {
plot.beginPath()
.moveTo(i*perW, 0)
.lineTo(i*perW, hBound)
.closePath()
.stroke();
}
var xpos, xpos2, ypos;
plot.setStrokeStyle('red')
.setLineWidth(4);
for (var i = 0; i < size; i++) {
ypos = hBound-(this.statisticalSample[i]-min)/dy*perH;
xpos = perW*(1+2*i);
if (i==0) {
plot.beginPath()
.moveTo(xpos, ypos);
}
else if (i < size-1) {
plot.lineTo(xpos, ypos);
}
else {
plot.lineTo(xpos, ypos)
.moveTo(xpos, ypos)
.closePath()
.stroke();
}
}
for (var i = 0; i < size; i++) {
ypos = hBound-(this.statisticalSample[i]-min)/dy*perH;
xpos = perW*(1+2*i);
xpos2 = xpos + 0.5*perW;
plot.setFillStyle('orange');
shape.fillCircle(xpos, ypos, 5);
plot.setFillStyle('blue');
plot.fillText(this.statisticalSample[i].toFixed(2), xpos, ypos-5, 100);
plot.setFillStyle('blue');
plot.setTextAlign('center');
if (i < lables) {
plot.fillText(lableArray[i], xpos2-0.5*perW,
hBound+30, 100);
}
}
plot.restore();
}
</span>
来个数据点少一点的：
<span style="font-size:18px;">>>>
试验次数--正面朝上次数---正面朝上频率
[ 5 , 3 , 0.6 ] ,
[ 15 , 9 , 0.6 ] ,
[ 25 , 16 , 0.64 ] ,
[ 35 , 15 , 0.42857142857142855 ] ,
[ 45 , 20 , 0.4444444444444444 ] ,
[ 55 , 31 , 0.5636363636363636 ] ,
[ 65 , 38 , 0.5846153846153846 ] ,
[ 75 , 44 , 0.5866666666666667 ] ,
[ 85 , 42 , 0.49411764705882355 ] ,
[ 95 , 50 , 0.5263157894736842 ] ,
if (1) {
var config = new PlotConfiguration();
config.init();
config.setPreference();
var r = 20;
//config.setSector(1,1,1,1);
//config.graphPaper2D(0, 0, r);
//config.axis2D(0, 0,180);
var array = [
[ 5 , 3 , 0.6 ] ,
[ 15 , 9 , 0.6 ] ,
[ 25 , 16 , 0.64 ] ,
[ 35 , 15 , 0.42857142857142855 ] ,
[ 45 , 20 , 0.4444444444444444 ] ,
[ 55 , 31 , 0.5636363636363636 ] ,
[ 65 , 38 , 0.5846153846153846 ] ,
[ 75 , 44 , 0.5866666666666667 ] ,
[ 85 , 42 , 0.49411764705882355 ] ,
[ 95 , 50 , 0.5263157894736842 ]
];
var len = array.length;
var stat = new Statistic();
var data = [], text = [];
//横坐标
for (var i = 0; i <len; i++) {
data.push(array[i][2]);
text.push(array[i][0].toFixed(0));
}
stat.init(data, '试验次数', '正面朝上频率', 1);
stat.lineGraph(text, 0, 0, 0, 1.0);
}</span>
假设尝试次数够多呢？
<span style="font-size:18px;">试验次数--正面朝上次数---正面朝上频率
[ 100 , 50 , 0.5 ] ,
[ 1100 , 548 , 0.49818181818181817 ] ,
[ 2100 , 1026 , 0.48857142857142855 ] ,
[ 3100 , 1522 , 0.49096774193548387 ] ,
[ 4100 , 2083 , 0.5080487804878049 ] ,
[ 5100 , 2613 , 0.5123529411764706 ] ,
[ 6100 , 3047 , 0.49950819672131147 ] ,
[ 7100 , 3522 , 0.496056338028169 ] ,
[ 8100 , 4083 , 0.5040740740740741 ] ,
[ 9100 , 4595 , 0.5049450549450549 ] ,
[ 10100 , 4960 , 0.4910891089108911 ] ,
if (1) {
var config = new PlotConfiguration();
config.init();
config.setPreference();
var r = 20;
//config.setSector(1,1,1,1);
//config.graphPaper2D(0, 0, r);
//config.axis2D(0, 0,180);
var array = [
[ 100 , 50 , 0.5 ] ,
[ 1100 , 548 , 0.49818181818181817 ] ,
[ 2100 , 1026 , 0.48857142857142855 ] ,
[ 3100 , 1522 , 0.49096774193548387 ] ,
[ 4100 , 2083 , 0.5080487804878049 ] ,
[ 5100 , 2613 , 0.5123529411764706 ] ,
[ 6100 , 3047 , 0.49950819672131147 ] ,
[ 7100 , 3522 , 0.496056338028169 ] ,
[ 8100 , 4083 , 0.5040740740740741 ] ,
[ 9100 , 4595 , 0.5049450549450549 ] ,
[ 10100 , 4960 , 0.4910891089108911 ]
];
var len = array.length;
var stat = new Statistic();
var data = [], text = [];
//横坐标
for (var i = 0; i <len; i++) {
data.push(array[i][2]);
text.push(array[i][0].toFixed(0));
}
stat.init(data, '试验次数', '正面朝上频率', 1);
stat.lineGraph(text, 0, 0, 0, 1.0);
}</span>
嗯，原来如此。
<span style="font-size:18px;">>>>
[0.14, 0.14, 1.0, 0.0, 0.6, 0.4]
>>> ================================ RESTART ================================
>>>
[0.23, 0.16, 1.0, 0.0, 0.5, 0.5]
>>> ================================ RESTART ================================
>>>
[0.14, 0.16, 1.0, 0.0, 0.51, 0.49]
#掷骰子
def dice():
N = 100;
count = [0]*6;
for i in range(N):
number = random.randint(1,6);
if number == 1:
count[0]+=1;
if number == 5:
count[1]+=1;
if number < 7:
count[2]+=1;
if number > 7:
count[3]+=1;
if number%2 == 0:
count[4]+=1;
if number%2 == 1:
count[5]+=1;
for i in range(len(count)):
count[i]/=N;
print(count);
</span><span style="font-size:18px;">>>>
两个骰子点数和为1的概率是0.0
两个骰子点数和为2的概率是0.02
两个骰子点数和为3的概率是0.07
两个骰子点数和为4的概率是0.13
两个骰子点数和为5的概率是0.08
两个骰子点数和为6的概率是0.11
两个骰子点数和为7的概率是0.14
两个骰子点数和为8的概率是0.13
两个骰子点数和为9的概率是0.13
两个骰子点数和为10的概率是0.1
两个骰子点数和为11的概率是0.04
两个骰子点数和为12的概率是0.05
#掷骰子, 两骰子点数和
def dice():
N = 100;
count = [0]*12;
for i in range(N):
number1 = random.randint(1,6);
number2 = random.randint(1,6);
count[number1+number2-1]+=1;
for i in range(len(count)):
count[i]/=N;
for i in range(len(count)):
print('两个骰子点数和为{0}的概率是{1}'.format(i+1, count[i]));</span>
假设实验次数够多，终于会是这种情况：
if (1) {
var config = new PlotConfiguration();
config.init();
config.setPreference();
var r = 20;
//config.setSector(1,1,1,1);
//config.graphPaper2D(0, 0, r);
//config.axis2D(0, 0,180);
var array = [0.0, 0.02, 0.05, 0.105, 0.105, 0.155, 0.175, 0.105, 0.12, 0.11, 0.045, 0.01];
var len = array.length;
var stat = new Statistic();
var data = [], text = [];
//横坐标
for (var i = 0; i <len; i++) {
data.push(array[i]);
text.push((i+1).toFixed(0));
}
stat.init(data, '点数和', '概率', 1);
stat.lineGraph(text, 0, 0, 0, 0.3);
}这种方法是不公平的，数据说明了一切。
>>> pow(1/6, 10);
1.6538171687920194e-08
[人叫板老师]点明了这‘出老千’的现象是真实存在的。
<span style="font-size:18px;">>>>
三天内有两天下雨的概率是0.4
未来三天的天气情况是
[[1, 9, 0], [6, 6, 3], [8, 5, 1], [4, 5, 3], [6, 6, 3], [1, 1, 0], [1, 4, 5], [6, 3, 6], [5, 0, 1], [1, 8, 6], [0, 2, 1], [3, 5, 4], [2, 6, 8], [1, 2, 8], [5, 3, 6], [4, 9, 1], [7, 4, 9], [0, 1, 8], [8, 9, 1], [3, 2, 7]]
>>>
三天内有两天下雨的概率是0.3
未来三天的天气情况是
[[3, 7, 9], [7, 5, 6], [5, 2, 0], [8, 8, 3], [6, 0, 3], [6, 1, 6], [5, 2, 8], [2, 2, 2], [1, 5, 5], [6, 6, 3], [0, 3, 2], [2, 2, 5], [6, 4, 7], [5, 7, 5], [1, 0, 2], [2, 6, 1], [3, 3, 4], [8, 8, 5], [2, 9, 1], [2, 0, 2]]
def weather():
N = 20;
array = [([0]*3) for i in range(N)];
for i in range(N):
for j in range(3):
array[i][j] = random.randint(0, 9);
count2 = 0;
for i in range(N):
count = 0;
for j in range(3):
if array[i][j]>=1 and array[i][j]<=4:
count+=1;
if count == 2:
count2+=1;
print('三天内有两天下雨的概率是{0}'.format(count2/N));
print('未来三天的天气情况是\n{0}'.format(array));
if __name__ == '__main__':
weather();</span>
<span style="font-size:18px;">>>>
试验次数--两个正面朝上次数---两个反面朝上次数
[ 100 , 38 , 35 ]
>>>
试验次数--两个正面朝上次数---两个反面朝上次数
[ 100 , 34 , 38 ]
>>>
试验次数--两个正面朝上次数---两个反面朝上次数
[ 100 , 32 , 40 ]
def coin():
result = [0]*2;
print('试验次数--两个正面朝上次数---两个反面朝上次数');
for i in range(100):
count = [0]*2;
for j in range(3):
number = random.randint(1, 10000);
if number%2 == 0:
count[0]+=1;
else:
count[1]+=1;
if (count[0] == 2):
result[0]+=1;
if (count[1] == 2):
result[1]+=1;
#result.append([i, count, count/i]);
print('[ 100',',',result[0],',',result[1], ']');
</span><span style="font-size:18px;">>>>
第1小题的概率是0.079
第2小题的概率是0.921
第3小题的概率是0.263
第4小题的概率是0.137
第5小题的概率是0.0
第6小题的概率是0.166
第7小题的概率是0.514
第8小题的概率是1.0
>>> ================================ RESTART ================================
>>>
第1小题的概率是0.093
第2小题的概率是0.907
第3小题的概率是0.242
第4小题的概率是0.153
第5小题的概率是0.0
第6小题的概率是0.153
第7小题的概率是0.515
第8小题的概率是1.0
def card():
#抽牌1000次
N = 1000;
#8个小题
result = [0]*8;
for i in range(N):
#52张牌
number = random.randint(1, 52);
#约定1-13 第一组红心，依次黑桃、方块、草花
#(1)
if number%13 == 7:
result[0]+=1;
#(2)
if number%13 != 7:
result[1]+=1;
#(3) 方片
if number//13==2:
result[2]+=1;
#(4) JQK
if number%13 > 10:
result[3]+=1;
#(5)
if number//13==0 and number//13 == 3:
result[4]+=1;
#(6)
if number%13 > 6 and number%13<9:
result[5]+=1;
#(7)
if (number//13)%2==0:
result[6]+=1;
#(8)
if (number//13)%2==0 or (number//13)%2==1:
result[7]+=1;
for i in range(len(result)):
print('第{0}小题的概率是{1}'.format(i+1, result[i]/N));
if __name__ == '__main__':
card();
</span>
<span style="font-size:18px;">>>>
[0.0, 0.35, 1.0]
>>> ================================ RESTART ================================
>>>
[0.0, 0.44, 1.0]
>>> ================================ RESTART ================================
>>>
[0.0, 0.4, 1.0]
def ball():
N = 100;
result = [0]*3;
for i in range(N):
#9个球编号1-9
number = random.randint(1, 9);
if number < 1:
result[0]+=1;
if number < 5:
result[1]+=1;
if number < 10:
result[2]+=1;
for i in range(len(result)):
result[i]/=N;
print(result);
if __name__ == '__main__':
ball();</span><span style="font-size:18px;">>>>
两个骰子点数和为1的概率是0.0
两个骰子点数和为2的概率是0.02
两个骰子点数和为3的概率是0.06
两个骰子点数和为4的概率是0.08
两个骰子点数和为5的概率是0.09
两个骰子点数和为6的概率是0.11
两个骰子点数和为7的概率是0.225
两个骰子点数和为8的概率是0.145
两个骰子点数和为9的概率是0.12
两个骰子点数和为10的概率是0.085
两个骰子点数和为11的概率是0.055
两个骰子点数和为12的概率是0.01
#掷骰子, 两骰子点数和
def dice():
N = 200;
count = [0]*12;
for i in range(N):
number1 = random.randint(1,6);
number2 = random.randint(1,6);
count[number1+number2-1]+=1;
for i in range(len(count)):
count[i]/=N;
for i in range(len(count)):
print('两个骰子点数和为{0}的概率是{1}'.format(i+1, count[i]));
if __name__ == '__main__':
dice();</span>
本节到此结束。欲知后事怎样，请看下回分解。
