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<script src="https://learn.mimmsy.com/learn-widget.js"></script>
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The whole widget is these files; the repository has their history.
widget.json
{
"name": "coin-entropy",
"title": "Biased coin entropy",
"version": 1,
"claim": "A coin's entropy H(p) = −p log₂ p − (1−p) log₂(1−p) peaks at exactly 1 bit per flip when p = ½ and falls to 0 as the coin becomes certain.",
"summary": "Drag P(heads) and watch a stream of flips: a fair coin is maximally unpredictable, a 95% coin produces a repetitive stream whose flips carry almost no information, and the readout follows the binary entropy curve.",
"topics": [
"information/entropy",
"math/probability"
],
"aliases": [
"binary entropy",
"Shannon entropy",
"H(p)",
"biased coin"
],
"params": {
"p": {
"type": "number",
"default": 0.5,
"min": 0,
"max": 1,
"label": "P(heads)"
}
},
"check": [
{
"q": "Which coin gives you the most information per flip, on average?",
"options": [
"P(heads) = 0.5",
"P(heads) = 0.99",
"P(heads) = 0.01",
"A two-headed coin"
],
"answer": 0,
"why": "The binary entropy curve peaks at p = ½, where a flip is least predictable and carries exactly 1 bit, while a two-headed coin's flips carry 0 bits."
},
{
"q": "A coin lands heads 95% of the time. Compared to a fair coin, its entropy per flip is…",
"options": [
"higher, because the rare outcome carries more information",
"the same, because there are still two outcomes",
"much lower, about 0.29 bits",
"exactly 0"
],
"answer": 2,
"why": "H(0.95) ≈ 0.29 bits: the rare tails carries 4.3 bits when it occurs, but it occurs too rarely to raise the average."
}
],
"capabilities": [],
"height": 490,
"requires": [
"surprise-bits"
],
"forkedFrom": null,
"authors": []
}
index.html
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Biased coin entropy</title>
<style>
:root {
color-scheme: light dark;
--bg: #f7f3ea; --bg-card: #efe9da; --border: #d8cfba; --text: #211d14; --text-dim: #6e6553;
--accent: #31597f; --accent2: #b04e1b; --hot: #a82433; --ok: #3d6b4f;
--ink-rgb: 33, 29, 20; --paper-rgb: 247, 243, 234; --accent-rgb: 49, 89, 127;
--accent2-rgb: 176, 78, 27; --hot-rgb: 168, 36, 51; --ok-rgb: 61, 107, 79;
--serif: "Iowan Old Style", "Palatino Linotype", Palatino, "Book Antiqua", Georgia, serif;
--mono: ui-monospace, "SF Mono", Menlo, Consolas, monospace;
}
@media (prefers-color-scheme: dark) {
:root {
--bg: #161410; --bg-card: #1e1b15; --border: #383225; --text: #e9e3d3; --text-dim: #9c917c;
--accent: #8fb8e0; --accent2: #dd9355; --hot: #df7a88; --ok: #82bd97;
--ink-rgb: 233, 227, 211; --paper-rgb: 22, 20, 16; --accent-rgb: 143, 184, 224;
--accent2-rgb: 221, 147, 85; --hot-rgb: 223, 122, 136; --ok-rgb: 130, 189, 151;
}
}
* { box-sizing: border-box; }
html, body { margin: 0; }
body { background: transparent; color: var(--text); font-family: var(--serif); font-size: 17px; line-height: 1.5; }
.widget { background: var(--bg-card); border: 1px solid var(--border); border-radius: 4px; padding: 22px; }
.control-row { display: flex; align-items: center; gap: 14px; flex-wrap: wrap; margin-bottom: 14px; }
.control-row label { font-size: 14px; color: var(--text-dim); white-space: nowrap; }
input[type="range"] { flex: 1; min-width: 160px; accent-color: var(--accent); margin: 0; }
.mono { font-family: var(--mono); font-variant-numeric: tabular-nums; }
.val { min-width: 4ch; font-size: 14px; }
.btn { font-family: var(--mono); font-size: 12.5px; color: var(--text); background: transparent; border: 1px solid var(--border); border-radius: 3px; padding: 6px 13px; cursor: pointer; transition: border-color .15s, color .15s; }
.btn:hover { border-color: var(--accent); color: var(--accent); }
.stream { font-family: var(--mono); font-size: 15px; letter-spacing: 3px; word-break: break-all; line-height: 1.5; height: 72px; overflow: hidden; background: var(--bg); border: 1px solid var(--border); border-radius: 10px; padding: 12px 14px; user-select: none; }
.stream .h { color: var(--accent); }
.stream .t { color: var(--accent2); }
.readout-row { display: flex; gap: 28px; flex-wrap: wrap; align-items: flex-end; margin: 14px 0; }
.readout .big { display: block; font-size: 34px; font-weight: 600; color: var(--accent); line-height: 1.1; min-width: 5ch; }
.readout.accent .big { color: var(--accent2); }
.readout small { color: var(--text-dim); font-size: 12.5px; }
.plot { display: block; width: 100%; height: 220px; margin-top: 8px; }
@media (max-width: 560px) { .readout .big { font-size: 27px; } }
</style>
</head>
<body>
<div class="widget">
<div class="control-row">
<label for="p">P(heads)</label>
<input type="range" id="p" min="0" max="100" value="50">
<span class="mono val" id="p-label">0.50</span>
<button id="reflip" class="btn">flip again</button>
</div>
<div id="stream" class="stream"></div>
<div class="readout-row">
<div class="readout">
<span class="big mono" id="h">1.00</span>
<small>H(p) · bits per flip</small>
</div>
<div class="readout accent">
<span class="big mono" id="i-heads">1.00</span>
<small>bits when heads · −log₂ p</small>
</div>
<div class="readout accent">
<span class="big mono" id="i-tails">1.00</span>
<small>bits when tails · −log₂(1−p)</small>
</div>
</div>
<canvas id="plot" class="plot"></canvas>
</div>
<script src="/w/_sdk/host.js?v=1"></script>
<script src="/w/_lib/info.js?v=1"></script>
<script src="widget.js?v=1"></script>
</body>
</html>
widget.js
/* Biased coin entropy: a stream of flips at P(heads) = p and the binary entropy curve H(p). */
(() => {
'use strict';
const $ = (id) => document.getElementById(id);
const { clamp, binaryH, readTheme, plotCurve, onResize } = window.Info;
let p = 0.5;
let host = null;
const surprise = (x) => (x > 0 ? (-Math.log2(x)).toFixed(2) : '∞');
function renderStream() {
const N = clamp(Math.floor(($('stream').clientWidth - 28) / 12.2) * 2, 24, 128);
let html = '';
for (let i = 0; i < N; i++) {
html += Math.random() < p ? '<span class="h">H</span>' : '<span class="t">T</span>';
}
$('stream').innerHTML = html;
}
function render() {
$('p-label').textContent = p.toFixed(2);
$('h').textContent = binaryH(p).toFixed(2);
$('i-heads').textContent = surprise(p);
$('i-tails').textContent = surprise(1 - p);
plotCurve($('plot'), {
fn: binaryH, xmin: 0, xmax: 1, ymax: 1.05, dotX: p, xlabel: 'P(heads)', ylabel: 'H bits',
});
}
function setP(v, fromSlider) {
p = clamp(v, 0, 1);
if (!fromSlider) $('p').value = Math.round(p * 100);
render();
renderStream();
}
const ping = (data) => { if (host && host.signal) host.signal('interaction', data); };
$('p').addEventListener('input', () => { setP($('p').value / 100, true); ping({ p }); });
$('reflip').addEventListener('click', () => { renderStream(); ping({ p }); });
onResize(render);
readTheme();
render();
renderStream();
LearnWidget.connect().then((h) => {
host = h;
readTheme();
if (typeof h.params.p === 'number') setP(h.params.p);
else render();
h.on('theme.changed', () => { readTheme(); render(); });
h.on('params.changed', (np) => { if (typeof np.p === 'number') setP(np.p); });
});
})();