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 - 独立事象と従属事象

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)、独立事象と従属事象、問29、30、31.を取り組んでみる。

29. 
    $\begin{array}{l}{P}_A\left(B\right)=\frac{2}{3}\\ P_A\left(B\right)\ne P\left(B\right)\\ P_A\left(C\right)=\frac{1}{3}\\ P_A\left(C\right)=P\left(C\right)\end{array}$
30. 1. 
       $\begin{array}{l}P\left(A\right)=\frac{13}{52}=\frac{1}{4}\\ P\left(B\right)=\frac{3·4}{52}=\frac{3}{13}\\ P\left(A\right)·P\left(B\right)=\frac{3}{52}\\ P\left(A\cap B\right)=\frac{3}{52}=\frac{3}{52}\\ 独立事象\end{array}$
    2. 
       $\begin{array}{l}P\left(A\right)=\frac{3}{6}=\frac{1}{2}\\ P\left(B\right)=\frac{2}{6}=\frac{1}{3}\\ P\left(A\right)·P\left(B\right)=\frac{1}{6}\\ P\left(A\cap B\right)=\frac{1}{6}\\ 独立事象\end{array}$
    3. 
       $\begin{array}{l}P\left(A\right)=\frac{1}{2}\\ P\left(B\right)=\frac{1}{2}\\ P\left(A\right)·P\left(B\right)=\frac{1}{4}\\ P\left(A\cap B\right)=\frac{1}{4}\\ 独立事象\end{array}$
    4. 
       $\begin{array}{l}P\left(A\right)=\frac{13·39}{52·51}=\frac{13}{4·17}=\frac{13}{68}\\ P\left(B\right)=\frac{13·12}{52·51}+\frac{39·13}{52·51}=\frac{4+13}{4·17}=\frac{1}{4}\\ P\left(A\cap B\right)=\frac{13·13}{52·51}=\frac{13}{4·51}=\frac{13}{204}\\ P\left(A\right)P\left(B\right)=\frac{13}{272}\\ P\left(A\cap B\right)\ne P\left(A\right)P\left(B\right)\end{array}$
31. 
    $\begin{array}{l}P\left(A\right)=p\\ P\left(A^{\prime }\right)=1-p\\ P\left(B\right)=q\\ P\left(A\cap B\right)=pq\\ P\left(A^{\prime }\right)P\left(B\right)=\left(1-p\right)q\\ P\left(A^{\prime }\cap B\right)\\ =P\left(B\right)-P\left(A\cap B\right)\\ =q-pq\\ =\left(1-p\right)q\\ \\ P\left(A^{\prime }\cap B\right)=P\left(A^{\prime }\right)P\left(B\right)\end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
ÂõûÊï∞: <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="sample29.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 suit = range(13).map(() => 0)
    .concat(range(13).map(() => 1))
    .concat(range(13).map(() => 2))
    .concat(range(13).map(() => 3)),
    spade = 0,
    club = 1,
    dia = 2,
    heart = 3;

let f = () => {
    let suit0 = suit.slice(),
        i = Math.floor(Math.random() * suit0.length),
        card1 = suit[i];

    suit0.splice(i, 1);
    i = Math.floor(Math.random() * suit0.length);
    let card2 = suit[i];

    return card1 === heart && card2 === spade;
};
let output = () => {
    p('30-4.');
    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 = 13 / 204;
    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();

30-4.
false
0.061
0.06372549019607843
0.0027254901960784284

回数: