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Tutoring abilities

Will a dog sprout wings and fly to school today, or will the sun rise in the morning?

Some things can never happen, while other things will definitely happen.

How do mathematicians measure whether something will happen or not?

**Probability** is a math word that means how likely something is to happen.

We measure chance on a straight line that starts at **0** and goes up to **1**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Probability Scale</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-tap-highlight-color: transparent;
      user-select: none;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 16px;
      gap: 16px;
    }

    .main-view {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
    }

    /* Visual Proportion Display */
    .visual-card {
      background: #ffffff;
      border-radius: 16px;
      padding: 14px 20px;
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.05);
    }

    .status-badge {
      font-size: 18px;
      font-weight: 700;
      letter-spacing: 0.3px;
      padding: 4px 14px;
      border-radius: 20px;
      transition: all 0.25s ease;
      text-align: center;
    }

    /* Marble / Chance visual bar */
    .chance-meter {
      width: 100%;
      height: 28px;
      background: #e2e8f0;
      border-radius: 14px;
      overflow: hidden;
      position: relative;
      border: 2px solid #cbd5e1;
    }

    .chance-fill {
      height: 100%;
      width: 50%;
      border-radius: 12px;
      transition: width 0.15s ease-out, background-color 0.25s ease;
    }

    /* Triple value readout */
    .values-row {
      display: grid;
      grid-template-columns: 1fr 1fr 1fr;
      width: 100%;
      gap: 8px;
      text-align: center;
    }

    .val-box {
      background: #f1f5f9;
      padding: 8px 4px;
      border-radius: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 48px;
    }

    .val-num {
      font-size: 17px;
      font-weight: 700;
      color: #0f172a;
      line-height: 1.1;
    }

    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 13px;
      line-height: 1;
    }

    .frac .top {
      border-bottom: 1.5px solid #0f172a;
      padding-bottom: 1px;
    }

    .frac .bot {
      padding-top: 1px;
    }

    /* Slider Container */
    .slider-area {
      width: 100%;
      background: #ffffff;
      border-radius: 16px;
      padding: 16px 20px 14px 20px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.05);
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    .range-wrap {
      position: relative;
      width: 100%;
      height: 36px;
      display: flex;
      align-items: center;
    }

    input[type="range"] {
      -webkit-appearance: none;
      appearance: none;
      width: 100%;
      height: 10px;
      border-radius: 5px;
      background: #e2e8f0;
      outline: none;
      margin: 0;
      cursor: pointer;
    }

    input[type="range"]::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 28px;
      height: 28px;
      border-radius: 50%;
      background: #22b7ff;
      border: 3px solid #ffffff;
      box-shadow: 0 2px 6px rgba(0,0,0,0.25);
      cursor: grab;
      transition: transform 0.1s ease;
    }

    input[type="range"]:active::-webkit-slider-thumb {
      cursor: grabbing;
      transform: scale(1.15);
    }

    input[type="range"]::-moz-range-thumb {
      width: 28px;
      height: 28px;
      border-radius: 50%;
      background: #22b7ff;
      border: 3px solid #ffffff;
      box-shadow: 0 2px 6px rgba(0,0,0,0.25);
      cursor: grab;
    }

    /* Milestone snap buttons */
    .milestones {
      display: flex;
      justify-content: space-between;
      width: 100%;
      position: relative;
    }

    .preset-btn {
      background: #f8fafc;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 6px 8px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      display: flex;
      flex-direction: column;
      align-items: center;
      min-width: 52px;
      transition: background-color 0.15s, border-color 0.15s, color 0.15s;
    }

    .preset-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
    }

    .preset-btn:active {
      filter: brightness(0.95);
    }

    .preset-sub {
      font-size: 11px;
      opacity: 0.85;
      font-weight: 500;
    }
  </style>
</head>
<body>

  <div class="main-view">
    
    <div class="visual-card">
      <div id="statusBadge" class="status-badge">Even Chance</div>
      
      
      <div class="chance-meter">
        <div id="chanceFill" class="chance-fill"></div>
      </div>

      
      <div class="values-row">
        <div class="val-box">
          <span id="percentText" class="val-num">50%</span>
        </div>
        <div class="val-box">
          <span id="fractionText" class="val-num">
            <span class="frac"><span class="top">1</span><span class="bot">2</span></span>
          </span>
        </div>
        <div class="val-box">
          <span id="decimalText" class="val-num">0.5</span>
        </div>
      </div>
    </div>

    
    <div class="slider-area">
      <div class="range-wrap">
        <input id="probSlider" type="range" min="0" max="100" value="50" step="1" aria-label="Probability scale">
      </div>

      
      <div class="milestones">
        <button class="preset-btn" data-val="0">
          <span>0</span>
          <span class="preset-sub">0%</span>
        </button>
        <button class="preset-btn" data-val="25">
          <span>¼</span>
          <span class="preset-sub">25%</span>
        </button>
        <button class="preset-btn active" data-val="50">
          <span>½</span>
          <span class="preset-sub">50%</span>
        </button>
        <button class="preset-btn" data-val="75">
          <span>¾</span>
          <span class="preset-sub">75%</span>
        </button>
        <button class="preset-btn" data-val="100">
          <span>1</span>
          <span class="preset-sub">100%</span>
        </button>
      </div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('probSlider');
    const badge = document.getElementById('statusBadge');
    const fill = document.getElementById('chanceFill');
    const pText = document.getElementById('percentText');
    const fText = document.getElementById('fractionText');
    const dText = document.getElementById('decimalText');
    const presetBtns = document.querySelectorAll('.preset-btn');

    function gcd(a, b) {
      return b === 0 ? a : gcd(b, a % b);
    }

    function update(val) {
      const v = Number(val);
      fill.style.width = v + '%';

      // Values text
      pText.textContent = v + '%';
      dText.textContent = (v / 100).toFixed(v % 10 === 0 && v !== 0 && v !== 100 ? 1 : (v === 0 || v === 100 ? 0 : 2));

      // Fraction formatting
      if (v === 0) {
        fText.textContent = '0';
      } else if (v === 100) {
        fText.textContent = '1';
      } else {
        const divisor = gcd(v, 100);
        const top = v / divisor;
        const bot = 100 / divisor;
        fText.innerHTML = `<span class="frac"><span class="top">${top}</span><span class="bot">${bot}</span></span>`;
      }

      // Qualitative Badge and Theme Colors
      let label = '';
      let bg = '';
      let color = '#ffffff';

      if (v === 0) {
        label = 'Impossible';
        bg = '#ef4444';
      } else if (v < 50) {
        label = 'Unlikely';
        bg = '#f59e0b';
      } else if (v === 50) {
        label = 'Even Chance';
        bg = '#22b7ff';
      } else if (v < 100) {
        label = 'Likely';
        bg = '#3b82f6';
      } else {
        label = 'Certain';
        bg = '#10b981';
      }

      badge.textContent = label;
      badge.style.backgroundColor = bg;
      badge.style.color = color;
      fill.style.backgroundColor = bg;

      // Update active milestone button
      presetBtns.forEach(btn => {
        btn.classList.toggle('active', Number(btn.dataset.val) === v);
      });
    }

    slider.addEventListener('input', (e) => update(e.target.value));

    presetBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        const val = Number(btn.dataset.val);
        slider.value = val;
        update(val);
      });
    });

    // Initial render
    update(50);
  </script>
</body>
</html>

What does it mean when a chance lands right in the middle?

When you flip a coin, you have an **equal chance** of landing on heads or tails.

We can write this middle chance as the fraction **$\frac{1}{2}$**, the decimal **$0.5$**, or the percentage **$50\%$**.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
    background: transparent;
    color: #1e2945;
    display: flex;
    justify-content: center;
    align-items: center;
    min-height: 100%;
    padding: 12px;
    user-select: none;
    -webkit-user-select: none;
  }

  .diagram-root {
    width: 100%;
    max-width: 520px;
    display: flex;
    flex-direction: column;
    gap: 16px;
  }

  /* Main Display Badge */
  .status-badge {
    align-self: center;
    background: #ffffff;
    border: 2px solid #e2e8f0;
    border-radius: 999px;
    padding: 6px 20px;
    font-weight: 700;
    font-size: 16px;
    letter-spacing: 0.5px;
    color: #0284c7;
    box-shadow: 0 2px 6px rgba(0,0,0,0.04);
    transition: color 0.25s ease, border-color 0.25s ease;
  }

  /* Values 3-Card Grid */
  .values-grid {
    display: grid;
    grid-template-columns: repeat(3, 1fr);
    gap: 10px;
  }

  .val-card {
    background: #ffffff;
    border: 1.5px solid #e2e8f0;
    border-radius: 12px;
    padding: 12px 6px;
    text-align: center;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 80px;
    box-shadow: 0 2px 5px rgba(0,0,0,0.03);
    transition: border-color 0.2s ease, box-shadow 0.2s ease;
  }

  .val-card.active-val {
    border-color: #22b7ff;
    box-shadow: 0 0 0 3px rgba(34, 183, 255, 0.15);
  }

  .val-label {
    font-size: 12px;
    text-transform: uppercase;
    font-weight: 600;
    color: #64748b;
    margin-bottom: 6px;
    letter-spacing: 0.3px;
  }

  .val-num {
    font-size: 20px;
    font-weight: 700;
    color: #0f172a;
    display: flex;
    align-items: center;
    justify-content: center;
    min-height: 32px;
  }

  /* Fraction display */
  .frac {
    display: inline-flex;
    flex-direction: column;
    vertical-align: middle;
    text-align: center;
    font-size: 17px;
    line-height: 1;
  }
  .frac-top {
    border-bottom: 1.5px solid #0f172a;
    padding-bottom: 2px;
  }
  .frac-bot {
    padding-top: 2px;
  }

  /* Interactive Track Container */
  .scale-container {
    position: relative;
    padding: 24px 10px 10px 10px;
  }

  .track-bar {
    position: relative;
    width: 100%;
    height: 14px;
    background: #e2e8f0;
    border-radius: 7px;
    cursor: pointer;
    overflow: hidden;
  }

  .track-fill {
    position: absolute;
    left: 0;
    top: 0;
    bottom: 0;
    width: 50%;
    background: linear-gradient(90deg, #94a3b8, #22b7ff 50%, #10b981);
    border-radius: 7px;
    transition: width 0.15s ease-out;
  }

  /* Native Range Input over Track */
  .range-slider {
    position: absolute;
    top: 20px;
    left: 10px;
    right: 10px;
    width: calc(100% - 20px);
    height: 22px;
    opacity: 0;
    cursor: pointer;
    z-index: 10;
    margin: 0;
  }

  /* Indicator Needle / Pointer */
  .pointer-marker {
    position: absolute;
    top: 8px;
    width: 24px;
    height: 24px;
    background: #0284c7;
    border: 3px solid #ffffff;
    border-radius: 50%;
    transform: translateX(-50%);
    box-shadow: 0 2px 6px rgba(0,0,0,0.25);
    pointer-events: none;
    transition: left 0.15s ease-out;
    z-index: 5;
  }

  /* Preset Anchor Buttons */
  .anchors-row {
    display: flex;
    justify-content: space-between;
    position: relative;
    margin-top: 14px;
  }

  .anchor-btn {
    background: #ffffff;
    border: 1.5px solid #cbd5e1;
    border-radius: 8px;
    padding: 6px 8px;
    font-size: 13px;
    font-weight: 600;
    color: #475569;
    cursor: pointer;
    transition: background-color 0.2s, border-color 0.2s, color 0.2s, box-shadow 0.2s;
    text-align: center;
  }

  .anchor-btn:hover {
    border-color: #22b7ff;
    color: #0284c7;
  }

  .anchor-btn.active {
    background: #0284c7;
    border-color: #0284c7;
    color: #ffffff;
    box-shadow: 0 2px 6px rgba(2, 132, 199, 0.3);
  }

  /* Visual Guide Scale */
  .ticks-row {
    display: flex;
    justify-content: space-between;
    padding: 0 4px;
    margin-top: 6px;
  }
  .tick-label {
    font-size: 11px;
    font-weight: 600;
    color: #94a3b8;
  }
</style>
</head>
<body>

<div class="diagram-root">
  
  <div class="status-badge" id="statusBadge">Even Chance</div>

  
  <div class="values-grid">
    <div class="val-card active-val">
      <div class="val-label">Fraction</div>
      <div class="val-num" id="fracVal">
        <span class="frac"><span class="frac-top">1</span><span class="frac-bot">2</span></span>
      </div>
    </div>
    <div class="val-card active-val">
      <div class="val-label">Decimal</div>
      <div class="val-num" id="decVal">0.5</div>
    </div>
    <div class="val-card active-val">
      <div class="val-label">Percentage</div>
      <div class="val-num" id="pctVal">50%</div>
    </div>
  </div>

  
  <div class="scale-container">
    <div class="track-bar" id="trackBar">
      <div class="track-fill" id="trackFill"></div>
    </div>
    <div class="pointer-marker" id="pointer"></div>
    <input type="range" min="0" max="100" value="50" step="1" class="range-slider" id="probRange" aria-label="Probability scale">
    
    <div class="ticks-row">
      <span class="tick-label">0</span>
      <span class="tick-label">0.25</span>
      <span class="tick-label">0.5</span>
      <span class="tick-label">0.75</span>
      <span class="tick-label">1</span>
    </div>
  </div>

  
  <div class="anchors-row">
    <button class="anchor-btn" data-val="0">Impossible</button>
    <button class="anchor-btn" data-val="25">Unlikely</button>
    <button class="anchor-btn active" data-val="50">Even</button>
    <button class="anchor-btn" data-val="75">Likely</button>
    <button class="anchor-btn" data-val="100">Certain</button>
  </div>
</div>

<script>
  const range = document.getElementById('probRange');
  const trackFill = document.getElementById('trackFill');
  const pointer = document.getElementById('pointer');
  const statusBadge = document.getElementById('statusBadge');
  const fracVal = document.getElementById('fracVal');
  const decVal = document.getElementById('decVal');
  const pctVal = document.getElementById('pctVal');
  const anchorBtns = document.querySelectorAll('.anchor-btn');

  function gcd(a, b) {
    return b ? gcd(b, a % b) : a;
  }

  function updateScale(val) {
    const v = Number(val);
    trackFill.style.width = v + '%';
    pointer.style.left = v + '%';

    // Percentage & Decimal
    pctVal.textContent = v + '%';
    decVal.textContent = (v / 100).toFixed(v % 10 === 0 && v !== 0 && v !== 100 ? 1 : 2).replace(/\.00$/, '').replace(/(\.\d)0$/, '$1');

    // Fraction
    if (v === 0) {
      fracVal.innerHTML = '0';
    } else if (v === 100) {
      fracVal.innerHTML = '1';
    } else {
      const g = gcd(v, 100);
      const top = v / g;
      const bot = 100 / g;
      fracVal.innerHTML = `<span class="frac"><span class="frac-top">${top}</span><span class="frac-bot">${bot}</span></span>`;
    }

    // Likelihood Label
    let label = '';
    if (v === 0) label = 'Impossible';
    else if (v < 50) label = 'Unlikely';
    else if (v === 50) label = 'Even Chance';
    else if (v < 100) label = 'Likely';
    else label = 'Certain';

    statusBadge.textContent = label;

    // Anchor active state
    anchorBtns.forEach(btn => {
      if (Number(btn.dataset.val) === v) {
        btn.classList.add('active');
      } else {
        btn.classList.remove('active');
      }
    });
  }

  range.addEventListener('input', (e) => {
    updateScale(e.target.value);
  });

  anchorBtns.forEach(btn => {
    btn.addEventListener('click', () => {
      const val = Number(btn.dataset.val);
      range.value = val;
      updateScale(val);
    });
  });

  // Initial render
  updateScale(50);
</script>

</body>
</html>

How do we write impossible and certain using these same three number styles?

An **impossible** event has a chance of **$0$**, which is also written as **$0.0$** or **$0\%$**.

A **certain** event will always happen, so its chance is **$1$**, which is **$1.0$** or **$100\%$**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-tap-highlight-color: transparent;
      user-select: none;
    }

    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .main-view {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      gap: 16px;
      align-items: center;
    }

    /* Chance Visual Card */
    .visual-card {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 12px;
      padding: 16px 20px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
    }

    .word-badge {
      font-size: 18px;
      font-weight: 700;
      letter-spacing: 0.5px;
      padding: 4px 14px;
      border-radius: 20px;
      background: #f0f7ff;
      color: #0070f3;
      transition: all 0.3s ease;
    }

    .meter-container {
      width: 100%;
      height: 24px;
      background: #eef2f6;
      border-radius: 12px;
      overflow: hidden;
      position: relative;
    }

    .meter-fill {
      height: 100%;
      width: 50%;
      background: linear-gradient(90deg, #22b7ff, #0070f3);
      border-radius: 12px;
      transition: width 0.3s ease, background 0.3s ease;
    }

    .values-row {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      width: 100%;
      gap: 8px;
    }

    .val-box {
      background: #f8f9fa;
      border: 1px solid #e6e6e6;
      border-radius: 8px;
      padding: 8px 4px;
      text-align: center;
      display: flex;
      flex-direction: column;
      gap: 2px;
    }

    .val-num {
      font-size: 17px;
      font-weight: 700;
      color: #1e2945;
    }

    .val-tag {
      font-size: 11px;
      color: #6b7280;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    /* Interactive Track / Buttons */
    .scale-wrapper {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .scale-buttons {
      display: grid;
      grid-template-columns: repeat(5, 1fr);
      gap: 6px;
      width: 100%;
    }

    .scale-btn {
      background: #ffffff;
      border: 1.5px solid #e6e6e6;
      border-radius: 8px;
      padding: 8px 2px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 2px;
      transition: background 0.2s ease, border-color 0.2s ease, box-shadow 0.2s ease;
    }

    .scale-btn:active {
      opacity: 0.8;
    }

    .scale-btn.active {
      background: #0070f3;
      border-color: #0070f3;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(0, 112, 243, 0.3);
    }

    .scale-btn.active .btn-sub {
      color: #e0f2fe;
    }

    .btn-sub {
      font-size: 11px;
      color: #6b7280;
      font-weight: normal;
    }

    .slider-container {
      width: 100%;
      padding: 0 4px;
    }

    input[type="range"] {
      width: 100%;
      -webkit-appearance: none;
      appearance: none;
      height: 6px;
      border-radius: 3px;
      background: #e2e8f0;
      outline: none;
      cursor: pointer;
    }

    input[type="range"]::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 22px;
      height: 22px;
      border-radius: 50%;
      background: #0070f3;
      border: 2px solid #ffffff;
      box-shadow: 0 1px 4px rgba(0, 0, 0, 0.25);
      cursor: pointer;
    }

    input[type="range"]::-moz-range-thumb {
      width: 22px;
      height: 22px;
      border-radius: 50%;
      background: #0070f3;
      border: 2px solid #ffffff;
      box-shadow: 0 1px 4px rgba(0, 0, 0, 0.25);
      cursor: pointer;
    }
  </style>
</head>
<body>

  <div class="main-view">
    
    <div class="visual-card">
      <div id="wordBadge" class="word-badge">Even Chance</div>
      
      <div class="meter-container">
        <div id="meterFill" class="meter-fill"></div>
      </div>

      <div class="values-row">
        <div class="val-box">
          <span id="fracVal" class="val-num">1/2</span>
          <span class="val-tag">Fraction</span>
        </div>
        <div class="val-box">
          <span id="decVal" class="val-num">0.5</span>
          <span class="val-tag">Decimal</span>
        </div>
        <div class="val-box">
          <span id="pctVal" class="val-num">50%</span>
          <span class="val-tag">Percent</span>
        </div>
      </div>
    </div>

    
    <div class="scale-wrapper">
      <div class="slider-container">
        <input type="range" id="probRange" min="0" max="4" step="1" value="2" aria-label="Probability Scale">
      </div>

      <div class="scale-buttons">
        <button class="scale-btn" data-idx="0">
          <span>0</span>
          <span class="btn-sub">0%</span>
        </button>
        <button class="scale-btn" data-idx="1">
          <span>1/4</span>
          <span class="btn-sub">25%</span>
        </button>
        <button class="scale-btn active" data-idx="2">
          <span>1/2</span>
          <span class="btn-sub">50%</span>
        </button>
        <button class="scale-btn" data-idx="3">
          <span>3/4</span>
          <span class="btn-sub">75%</span>
        </button>
        <button class="scale-btn" data-idx="4">
          <span>1</span>
          <span class="btn-sub">100%</span>
        </button>
      </div>
    </div>
  </div>

  <script>
    const data = [
      { word: "Impossible", fill: "0%", frac: "0", dec: "0.0", pct: "0%" },
      { word: "Unlikely", fill: "25%", frac: "1/4", dec: "0.25", pct: "25%" },
      { word: "Even Chance", fill: "50%", frac: "1/2", dec: "0.5", pct: "50%" },
      { word: "Likely", fill: "75%", frac: "3/4", dec: "0.75", pct: "75%" },
      { word: "Certain", fill: "100%", frac: "1", dec: "1.0", pct: "100%" }
    ];

    const wordBadge = document.getElementById("wordBadge");
    const meterFill = document.getElementById("meterFill");
    const fracVal = document.getElementById("fracVal");
    const decVal = document.getElementById("decVal");
    const pctVal = document.getElementById("pctVal");
    const probRange = document.getElementById("probRange");
    const buttons = document.querySelectorAll(".scale-btn");

    function setStep(idx) {
      const cur = data[idx];
      wordBadge.textContent = cur.word;
      meterFill.style.width = cur.fill;
      fracVal.textContent = cur.frac;
      decVal.textContent = cur.dec;
      pctVal.textContent = cur.pct;
      probRange.value = idx;

      buttons.forEach((btn, i) => {
        btn.classList.toggle("active", i === idx);
      });
    }

    buttons.forEach((btn, idx) => {
      btn.addEventListener("click", () => setStep(idx));
    });

    probRange.addEventListener("input", (e) => {
      setStep(parseInt(e.target.value, 10));
    });
  </script>
</body>
</html>

What happens if a game is likely, but not totally certain?

Imagine a jar with $3$ blue marbles and $1$ red marble.

Picking a blue marble is **likely**, so its probability is **$\frac{3}{4}$**, which is **$0.75$** or **$75\%$**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Probability Scale</title>
  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css">
  <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }
    .main-container {
      width: 100%;
      max-width: 580px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }
    
    /* Top display card */
    .display-card {
      background: #ffffff;
      border: 2px solid #e2e8f0;
      border-radius: 14px;
      padding: 12px 16px;
      display: grid;
      grid-template-columns: 1fr 1fr;
      align-items: center;
      gap: 12px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.04);
    }
    .visual-gauge {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
    }
    .pie-wrap {
      position: relative;
      width: 80px;
      height: 80px;
    }
    svg.pie-svg {
      width: 80px;
      height: 80px;
      transform: rotate(-90deg);
      border-radius: 50%;
    }
    .pie-bg {
      fill: #f1f5f9;
    }
    .pie-fill {
      fill: #22b7ff;
      transition: stroke-dasharray 0.3s ease;
    }
    .stats-col {
      display: flex;
      flex-direction: column;
      gap: 6px;
    }
    .word-badge {
      display: inline-block;
      align-self: flex-start;
      font-size: 14px;
      font-weight: 700;
      padding: 3px 10px;
      border-radius: 20px;
      background: #e0f2fe;
      color: #0369a1;
      transition: all 0.25s ease;
    }
    .values-row {
      display: flex;
      align-items: baseline;
      gap: 10px;
      font-size: 15px;
      color: #334155;
    }
    .val-item {
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .val-item strong {
      color: #1e2945;
      font-size: 16px;
    }

    /* Slider track section */
    .scale-box {
      position: relative;
      padding: 20px 10px 10px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 14px;
    }
    .slider-wrap {
      position: relative;
      width: 100%;
      height: 24px;
      display: flex;
      align-items: center;
    }
    input[type=range] {
      -webkit-appearance: none;
      width: 100%;
      height: 10px;
      border-radius: 5px;
      background: linear-gradient(to right, #94a3b8 0%, #22b7ff 50%, #10b981 100%);
      outline: none;
      cursor: pointer;
    }
    input[type=range]::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 26px;
      height: 26px;
      border-radius: 50%;
      background: #ffffff;
      border: 3px solid #0284c7;
      box-shadow: 0 2px 6px rgba(0,0,0,0.25);
      cursor: grab;
      transition: transform 0.1s;
    }
    input[type=range]:active::-webkit-slider-thumb {
      cursor: grabbing;
      transform: scale(1.15);
    }
    input[type=range]::-moz-range-thumb {
      width: 24px;
      height: 24px;
      border-radius: 50%;
      background: #ffffff;
      border: 3px solid #0284c7;
      box-shadow: 0 2px 6px rgba(0,0,0,0.25);
      cursor: grab;
    }

    /* Benchmark buttons */
    .benchmarks {
      display: grid;
      grid-template-columns: repeat(5, 1fr);
      gap: 4px;
      margin-top: 14px;
    }
    .bench-btn {
      background: #ffffff;
      border: 1.5px solid #cbd5e1;
      border-radius: 8px;
      padding: 6px 2px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 2px;
      cursor: pointer;
      font-size: 11px;
      color: #475569;
      transition: all 0.2s ease;
      min-height: 48px;
    }
    .bench-btn.active {
      border-color: #0284c7;
      background: #e0f2fe;
      color: #0369a1;
      box-shadow: 0 0 0 1px #0284c7;
    }
    .bench-btn span.katex-wrap {
      font-size: 13px;
      font-weight: 600;
    }
    .bench-word {
      font-size: 10px;
      text-transform: capitalize;
      line-height: 1.1;
      text-align: center;
    }
    
    @media (max-width: 400px) {
      .bench-word {
        font-size: 9px;
      }
      .display-card {
        padding: 8px 12px;
      }
      .pie-wrap {
        width: 68px;
        height: 68px;
      }
      svg.pie-svg {
        width: 68px;
        height: 68px;
      }
    }
  </style>
</head>
<body>
  <div class="main-container">
    
    <div class="display-card">
      <div class="visual-gauge">
        <div class="pie-wrap">
          <svg class="pie-svg" viewBox="0 0 36 36">
            <path class="pie-bg" d="M18 2.0845 a 15.9155 15.9155 0 0 1 0 31.831 a 15.9155 15.9155 0 0 1 0 -31.831" fill="none" stroke="#e2e8f0" stroke-width="6"/>
            <path id="piePath" class="pie-fill" stroke-dasharray="75, 100" d="M18 2.0845 a 15.9155 15.9155 0 0 1 0 31.831 a 15.9155 15.9155 0 0 1 0 -31.831" fill="none" stroke="#0284c7" stroke-width="6" stroke-linecap="round"/>
          </svg>
        </div>
      </div>

      <div class="stats-col">
        <div id="wordBadge" class="word-badge">Likely</div>
        <div class="values-row">
          <div class="val-item" id="fracVal"></div>
          <div class="val-item">• <strong id="decVal">0.75</strong></div>
          <div class="val-item">• <strong id="pctVal">75%</strong></div>
        </div>
      </div>
    </div>

    
    <div class="scale-box">
      <div class="slider-wrap">
        <input type="range" id="probRange" min="0" max="100" value="75" step="1">
      </div>

      
      <div class="benchmarks">
        <button class="bench-btn" data-val="0">
          <span class="katex-wrap" data-tex="0">0</span>
          <span class="bench-word">Impossible</span>
        </button>
        <button class="bench-btn" data-val="25">
          <span class="katex-wrap" data-tex="\frac{1}{4}">1/4</span>
          <span class="bench-word">Unlikely</span>
        </button>
        <button class="bench-btn" data-val="50">
          <span class="katex-wrap" data-tex="\frac{1}{2}">1/2</span>
          <span class="bench-word">Evens</span>
        </button>
        <button class="bench-btn active" data-val="75">
          <span class="katex-wrap" data-tex="\frac{3}{4}">3/4</span>
          <span class="bench-word">Likely</span>
        </button>
        <button class="bench-btn" data-val="100">
          <span class="katex-wrap" data-tex="1">1</span>
          <span class="bench-word">Certain</span>
        </button>
      </div>
    </div>
  </div>

  <script>
    document.addEventListener('DOMContentLoaded', () => {
      const slider = document.getElementById('probRange');
      const piePath = document.getElementById('piePath');
      const wordBadge = document.getElementById('wordBadge');
      const decVal = document.getElementById('decVal');
      const pctVal = document.getElementById('pctVal');
      const fracVal = document.getElementById('fracVal');
      const benchBtns = document.querySelectorAll('.bench-btn');

      function renderTex() {
        if (window.katex) {
          document.querySelectorAll('.katex-wrap').forEach(el => {
            const tex = el.getAttribute('data-tex');
            if (tex) katex.render(tex, el, { throwOnError: false });
          });
        }
      }

      function getFractionTex(val) {
        if (val === 0) return "0";
        if (val === 100) return "1";
        if (val === 50) return "\\frac{1}{2}";
        if (val === 25) return "\\frac{1}{4}";
        if (val === 75) return "\\frac{3}{4}";
        if (val % 20 === 0) return `\\frac{${val/20}}{5}`;
        if (val % 10 === 0) return `\\frac{${val/10}}{10}`;
        return `\\frac{${val}}{100}`;
      }

      function updateState(val) {
        val = parseInt(val, 10);
        slider.value = val;

        // Gauge arc
        piePath.setAttribute('stroke-dasharray', `${val}, 100`);

        // Decimal & Percentage
        const dec = (val / 100).toFixed(val % 10 === 0 && val !== 0 && val !== 100 ? 1 : 2);
        decVal.textContent = val === 0 ? "0" : (val === 100 ? "1.0" : dec);
        pctVal.textContent = `${val}%`;

        // Fraction rendering
        const tex = getFractionTex(val);
        if (window.katex) {
          katex.render(tex, fracVal, { throwOnError: false });
        } else {
          fracVal.textContent = tex;
        }

        // Word Descriptor
        let word = "Even Chance";
        let colorBg = "#e0f2fe";
        let colorText = "#0369a1";

        if (val === 0) {
          word = "Impossible";
          colorBg = "#f1f5f9";
          colorText = "#475569";
        } else if (val < 45) {
          word = "Unlikely";
          colorBg = "#fef3c7";
          colorText = "#b45309";
        } else if (val <= 55) {
          word = "Even Chance";
          colorBg = "#e0f2fe";
          colorText = "#0369a1";
        } else if (val < 100) {
          word = "Likely";
          colorBg = "#dcfce7";
          colorText = "#15803d";
        } else {
          word = "Certain";
          colorBg = "#bbf7d0";
          colorText = "#166534";
        }

        wordBadge.textContent = word;
        wordBadge.style.backgroundColor = colorBg;
        wordBadge.style.color = colorText;

        // Highlight matching preset button
        benchBtns.forEach(btn => {
          if (parseInt(btn.getAttribute('data-val'), 10) === val) {
            btn.classList.add('active');
          } else {
            btn.classList.remove('active');
          }
        });
      }

      slider.addEventListener('input', (e) => {
        updateState(e.target.value);
      });

      benchBtns.forEach(btn => {
        btn.addEventListener('click', () => {
          const v = btn.getAttribute('data-val');
          updateState(v);
        });
      });

      // Initial render check for katex
      const checkKatex = setInterval(() => {
        if (window.katex) {
          clearInterval(checkKatex);
          renderTex();
          updateState(75);
        }
      }, 50);

      // Fallback init
      updateState(75);
    });
  </script>
</body>
</html>

Next time you roll a die or check the weather, see if you can point to where its chance lives on the $0$ to $1$ scale!

math

Understanding the Probability Scale from 0 to 1 and Percentages. Express probabilities on a continuum from 0 (impossible) to 1 (certain) using fractions, decimals, and percentages.

Third Level Maths & Numeracy: Probability and Chance (MNU 3-22a)

Probability on a Number Line and Basic Theoretical Probability

Matching Fractions, Decimals, Percentages and Probability Scales

Ensure students bridge concrete physical manipulatives (spinners, multi-sided dice, counters) with the abstract number line continuum. Emphasise equivalence across representations ($rac{a}{b} \leftrightarrow 0.xx \leftrightarrow xx\%$) and contextualise mathematics within responsible decision-making by pairing numerical odds with real-world consequence analysis.

Understanding the Probability Scale from 0 to 1 and Percentages

The Probability Line

The Middle Chance