数学 - JavaScript - 確からしさをみる - 確率 – 条件つき確率と確率の乗法定理 - 重複試行の確率

数学読本〈4〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

数学読本〈4〉数列の極限，順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第16章(確からしさをみる - 確率)、16.2(条件つき確率と確率の乗法定理)、重複試行の確率、問32、33、34、35、36.を取り組んでみる。

32. 
    $\begin{array}{l}p=0.5\\ 0.0625\left(1+2+2.5+20·0.125\right)\\ =0.0625·8\\ =0.5\\ \\ p=0.3\\ 0.0081\left(1+2.8+4.9+20*0.343\right)\\ =0.126036\\ 0.126\end{array}$
33. 1. 
       $\begin{array}{l}\left(\begin{array}{c}10\\ 4\end{array}\right)\frac{1}{2^{10}}\\ =\frac{10·9·8·7}{4·3·2}·\frac{1}{2^{10}}\\ =10·3·7·\frac{1}{2^{10}}\\ =\frac{105}{512}\end{array}$
    2. 
       $\begin{array}{l}\left(\begin{array}{c}10\\ 5\end{array}\right)\frac{1}{2^{10}}\\ =\frac{10·9·8·7·6}{5·4·3·2}·\frac{1}{2^{10}}\\ =\frac{9·7}{2^9}\\ =\frac{63}{512}\end{array}$
    3. 
       $\begin{array}{l}\left(\begin{array}{c}10\\ 0\end{array}\right)\frac{1}{2^{10}}+\left(\begin{array}{c}10\\ 1\end{array}\right)\frac{1}{2^{10}}+\left(\begin{array}{c}10\\ 2\end{array}\right)\frac{1}{2^{10}}+\left(\begin{array}{c}10\\ 3\end{array}\right)\frac{1}{2^{10}}\\ =\frac{1}{2^{10}}\left(1+10+45+120\right)\\ =\frac{176}{1024}\\ =\frac{11}{64}\end{array}$
34. 1. 
       $\begin{array}{l}\left(\begin{array}{c}5\\ 2\end{array}\right){\left(\frac{2}{6}\right)}^2·{\left(\frac{4}{6}\right)}^3\\ =\frac{10·8}{3^5}\\ =\frac{80}{243}\end{array}$
    2. 
       $\begin{array}{l}1-\left(\left(\begin{array}{c}5\\ 0\end{array}\right){\left(\frac{2}{6}\right)}^0·{\left(\frac{4}{6}\right)}^5+\left(\begin{array}{c}5\\ 1\end{array}\right){\left(\frac{2}{6}\right)}^1·{\left(\frac{4}{6}\right)}^4\right)\\ =1-\frac{32+5·16}{3^5}\\ =1-\frac{32+80}{3^5}\\ =1-\frac{112}{243}\\ =\frac{131}{243}\end{array}$
35. 1. 
       $\begin{array}{l}\left(\begin{array}{c}8\\ 4\end{array}\right){\left(\frac{1}{2}\right)}^8\\ =\frac{8·7·6·5}{4·3·2}·\frac{1}{2^8}\\ =\frac{35}{128}\end{array}$
    2. 
       $\begin{array}{l}\left(\begin{array}{c}8\\ 3\end{array}\right){\left(\frac{1}{2}\right)}^8\\ =\frac{8·7·6}{3·2}·\frac{1}{2^8}\\ =\frac{7}{32}\end{array}$
    3. 
       $\begin{array}{l}\left(\begin{array}{c}8\\ 2\end{array}\right){\left(\frac{1}{2}\right)}^8\\ =\frac{8·7}{2}·\frac{1}{2^8}\\ =\frac{7}{64}\end{array}$
36. 
    $\begin{array}{l}\left(\begin{array}{c}n\\ 0\end{array}\right)\frac{1}{2^n}+\left(\begin{array}{c}n\\ 2\end{array}\right)\frac{1}{2^n}+···\\ =\frac{2^{n-1}}{2^n}\\ =\frac{1}{2}\\ \\ 1-\frac{1}{2}=\frac{1}{2}\end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
n = <input id="n0" type="number" min="1" step="1" value="1000">
<br>
<button id="run0">run</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>
<script src="sample32.js"></script>    

JavaScript

let pre0 = document.querySelector('#output0'),
    input_n = document.querySelector('#n0'),
    btn0 = document.querySelector('#run0'),
    btn1 = document.querySelector('#clear0'),
    div0 = document.querySelector('#graph0'),
    width = 600,
    height = 600,
    padding = 50,
    p = (x) => pre0.textContent += x + '\n';

let range = (n) => {
    let result = [];
    for (let i = 0; i < n; i += 1) {
        result.push(i);
    }
    return result;
};

let output = () => {
    p('36-4. 表');
    let n = parseInt(input_n.value, 10),
        points = [];

    points = range(n).map((i) => {
        return [i + 1,
                range(i + 1)
                .map(() => Math.floor(Math.random() * 2))
                .filter((b) => b === 0)
                .length / (i + 1)]
    });
    
    let t = points[points.length - 1][1],
        result = 1 / 2;
    p(t ===  result);
    p(t);
    p(result);
    p(Math.abs(t - result));

    let xscale = d3.scaleLinear()
        .domain([1, n])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([0, 1])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', 1)
        .attr('fill', 'green');

    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
};

input_n.onchange = output;
btn0.onclick = output;
btn1.onclick = () => pre0.textContent = '';

output();

36-4. 表
false
0.493
0.5
0.007000000000000006

n =