数学 - JavaScript - 確からしさをみる - 確率 – 確率とその基本性質 - 確率の計算(極限)

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)、16.1(確率とその基本性質)、確率の計算、問6、7、8、9、10、11.を取り組んでみる。

6. 
   $1-\frac{1}{2^5}=\frac{31}{32}$
7. 1. 
      $\frac{\left(\begin{array}{c}5\\ 2\end{array}\right)}{\left(\begin{array}{c}20\\ 2\end{array}\right)}=\frac{10}{190}=\frac{1}{19}$
   2. 
      $1-\frac{\left(\begin{array}{c}15\\ 2\end{array}\right)}{\left(\begin{array}{c}20\\ 2\end{array}\right)}=1-\frac{105}{190}=1-\frac{21}{38}=\frac{17}{38}$
   3. 
      $1-\frac{\left(\begin{array}{c}15\\ 3\end{array}\right)}{\left(\begin{array}{c}20\\ 3\end{array}\right)}=1-\frac{15·14·13}{20·19·18}=1-\frac{7·13}{4·19·3}=1-\frac{91}{228}=\frac{137}{228}$
   4. 
      $1-\frac{\left(\begin{array}{c}15\\ 3\end{array}\right)+\left(\begin{array}{c}5\\ 3\end{array}\right)}{\left(\begin{array}{c}20\\ 3\end{array}\right)}=1-\frac{15·14·13+5·4·3}{20·19·18}=1-\frac{14·13+4}{4·19·6}=1-\frac{93}{12·19}=\frac{135}{228}=\frac{45}{76}$
8. 1. 
      $1-\frac{4^3}{6^3}=1-\frac{2^3}{3^3}=1-\frac{8}{27}=\frac{19}{27}$
   2. 
      $1-\frac{2^3}{6^3}=1-\frac{1}{27}=\frac{26}{27}$
   3. 
      $\left(1-\frac{4^3}{6^3}\right)+\left(1-\frac{4^3}{6^3}\right)-\left(1-\frac{2^3}{6^3}\right)=\frac{12}{27}=\frac{4}{9}$
9. 1. 
      $\frac{\left(\begin{array}{c}3\\ 2\end{array}\right)+\left(\begin{array}{c}4\\ 2\end{array}\right)+\left(\begin{array}{c}5\\ 2\end{array}\right)}{\left(\begin{array}{c}12\\ 2\end{array}\right)}=\frac{3·2+4·3+5·4}{12·11}=\frac{38}{132}=\frac{19}{66}$
   2. 
      $1-\frac{\left(\begin{array}{c}7\\ 2\end{array}\right)}{\left(\begin{array}{c}12\\ 2\end{array}\right)}=1-\frac{7·6}{12·11}=1-\frac{7}{22}=\frac{15}{22}$
10. 1. 
       $\begin{array}{l}{p}_n=\frac{\left(\begin{array}{c}n\\ 2\end{array}\right)+\left(\begin{array}{c}n\\ 2\end{array}\right)}{\left(\begin{array}{c}2n\\ 2\end{array}\right)}=\frac{2n\left(n-1\right)}{2n\left(2n-1\right)}=\frac{n-1}{2n-1}\\ q_n=1-\frac{n-1}{2n-1}=\frac{2n-1-n+1}{2n-1}=\frac{n}{2n-1}\end{array}$
    2. 
       $p_n<q_n$
    3. 
       $\begin{array}{l}\underset{n\to \infty }{\lim }{p}_n=\underset{n\to \infty }{\lim }\frac{n-1}{2n-1}=\frac{1}{2}\\ \underset{n\to \infty }{\lim }{q}_n=\underset{n\to \infty }{\lim }\frac{n}{2n-1}=\frac{1}{2}\end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
n = <input id="n0" type="number" min="2" step="1" value="100">
<br>
<button id="draw0">draw</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="sample6.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    input0 = document.querySelector('#n0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 20,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    p = (x) => pre0.textContent += x + '\n',
    pn = (n) => (n - 1) / (2 * n - 1),
    qn = (n) => n / (2 * n - 1);

let draw = () => {
    let points = [],
        n = parseInt(input0.value, 10);

    for (let i = 2; i <= n; i += 1) {
        points.push([i, pn(i)]);
    }
    for (let i = 2; i <= n; i += 1) {
        points.push([i, qn(i)]);
    }
    let xscale = d3.scaleLinear()
        .domain([2, n])
        .range([padding, width - padding]);
    let ys = points.map((a) => a[1]);
    let yscale = d3.scaleLinear()
        .domain([Math.min(...ys), Math.max(...ys)])
        .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);

    let t = points.length / 2;
    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', (d, i) => i < t ? 'green' : 'blue');

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

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p('n, pn, qn, difference');
    for (let i = 2; i <= n; i += 1) {
        let a = pn(i),
            b = qn(i);
        p(`${i}, ${a}, ${b}, ${b - a}`);
    }
}

input0.onchange = draw;
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';

draw();

n =