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 - いくつかの例(組み合わせ)

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)、いくつかの例、問24、25、26.を取り組んでみる。

24. 
    $\begin{array}{l}\frac{1}{6}·\frac{10}{100}+\frac{1}{6}·\frac{20}{100}+\frac{1}{6}·\frac{30}{100}+\frac{1}{6}·\frac{40}{100}+\frac{1}{6}·\frac{50}{100}+\frac{1}{6}·\frac{60}{100}\\ =\frac{21}{60}\\ =\frac{7}{20}\end{array}$
25. 1. 
       $\begin{array}{l}\frac{1}{2}·\frac{3}{5}+\frac{1}{2}·\frac{3}{7}\\ =\frac{3}{10}+\frac{3}{14}\\ =\frac{21+15}{70}\\ =\frac{36}{70}\\ =\frac{18}{35}\end{array}$
    2. 
       $\frac{\frac{1}{2}·\frac{3}{5}}{\frac{18}{35}}=\frac{3·35}{18·10}=\frac{1·7}{6·2}=\frac{7}{12}$
    3. 
       $\frac{\frac{1}{2}·\frac{3}{7}}{\frac{18}{35}}=\frac{3·35}{18·14}=\frac{5}{6·2}=\frac{5}{12}$
26. 1. 
       $\left(\begin{array}{c}4\\ 2\end{array}\right){\left(\frac{4}{8}\right)}^4=6·\frac{1}{2^4}=\frac{3}{8}$
    2. 
       $\begin{array}{l}\frac{4·4·4·4+\left(\begin{array}{c}4\\ 2\end{array}\right)\left(\begin{array}{c}4\\ 2\end{array}\right)2}{\left(\begin{array}{c}8\\ 2\end{array}\right)·\left(\begin{array}{c}8\\ 2\end{array}\right)}\\ =\frac{4^4+6{·}^{2}2}{28·28}\\ =\frac{2·4^2+3·3}{14·7}\\ =\frac{32+9}{98}\\ =\frac{41}{98}\end{array}$
    3. 
       $\frac{\left(\begin{array}{c}4\\ 2\end{array}\right)\left(\begin{array}{c}4\\ 2\end{array}\right)}{\left(\begin{array}{c}8\\ 4\end{array}\right)}=\frac{36}{70}=\frac{18}{35}$
27. 
    

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
回数: <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="sample24.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 balls = [0, 0, 0, 0, 1, 1, 1, 1],
    len = balls.length,
    white = 0,
    black = 1;

let f = () => {
    let balls0 = balls.slice(),
        i = Math.floor(Math.random() * balls0.length),
        b1 = balls0[i];

    balls0.splice(i, 1);
    let b2 = balls0[Math.floor(Math.random() * balls0.length)];
    balls0.push(b1);
    balls0.push(b2);
    
    i = Math.floor(Math.random() * balls0.length);
    let b3 = balls0[i];
    balls0.splice(i, 1);
    let b4 = balls0[Math.floor(Math.random() * balls0.length)];

    return [b1, b2, b3, b4].filter((b) => b === white).length === 2;
};
let output = () => {
    p('26-2.');
    let n = parseInt(input_n.value, 10),
        results = range(n).map(() => f()),
        points = [];

    points = range(n).map((i) => {
        return [i + 1,
                range(i + 1)
                .map(() => f())
                .filter((b) => b)
                .length / (i + 1)]
    });
    
    let t = points[points.length - 1][1],
        result = 41 / 98;
    p(t ===  result);
    p(t);
    p(result);
    p(Math.abs(t - result));

    let xscale = d3.scaleLinear()
        .domain([1, 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);

    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();

回数: