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Tutoring abilities

Imagine a carnival booth where a giant wheel has two equal slices: one red and one blue. It costs $1 to spin, and if you land on red, you win $2!

If you play this game all day, do you expect to make money, lose money, or break even?

The **expected value** is the average amount of money you win or lose each time you play a game.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }

    .main-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 16px;
      width: 100%;
      max-width: 520px;
      align-items: center;
    }

    @media (max-width: 440px) {
      .main-grid {
        grid-template-columns: 1fr;
        gap: 12px;
      }
    }

    .spinner-area {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }

    .spinner-container {
      position: relative;
      width: 150px;
      height: 150px;
      border-radius: 50%;
      box-shadow: 0 4px 12px rgba(0,0,0,0.08);
    }

    .wheel {
      width: 100%;
      height: 100%;
      border-radius: 50%;
      background: conic-gradient(#ef4444 0deg 180deg, #3b82f6 180deg 360deg);
      position: relative;
      overflow: hidden;
      border: 3px solid #ffffff;
    }

    .sector-label {
      position: absolute;
      font-weight: 700;
      font-size: 15px;
      color: #ffffff;
      text-shadow: 0 1px 2px rgba(0,0,0,0.3);
    }

    .label-red {
      top: 50%;
      right: 20px;
      transform: translateY(-50%);
    }

    .label-blue {
      top: 50%;
      left: 20px;
      transform: translateY(-50%);
    }

    .needle {
      position: absolute;
      top: 50%;
      left: 50%;
      width: 4px;
      height: 58px;
      background: #1e2945;
      transform-origin: 50% 90%;
      transform: translate(-50%, -90%) rotate(0deg);
      transition: transform 0.8s cubic-bezier(0.15, 0.9, 0.3, 1);
      border-radius: 2px;
      z-index: 5;
    }

    .needle::before {
      content: '';
      position: absolute;
      top: -6px;
      left: 50%;
      transform: translateX(-50%);
      border-left: 6px solid transparent;
      border-right: 6px solid transparent;
      border-bottom: 10px solid #1e2945;
    }

    .pin {
      position: absolute;
      top: 50%;
      left: 50%;
      width: 14px;
      height: 14px;
      background: #ffffff;
      border: 3px solid #1e2945;
      border-radius: 50%;
      transform: translate(-50%, -50%);
      z-index: 6;
    }

    .calc-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 14px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    .calc-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      padding: 6px 8px;
      border-radius: 8px;
      font-size: 14px;
    }

    .row-red { background: #fef2f2; color: #991b1b; }
    .row-blue { background: #eff6ff; color: #1e40af; }

    .badge {
      display: inline-block;
      width: 8px;
      height: 8px;
      border-radius: 50%;
      margin-right: 6px;
    }

    .badge-red { background: #ef4444; }
    .badge-blue { background: #3b82f6; }

    .calc-total {
      border-top: 2px dashed #e5e7eb;
      padding-top: 8px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      font-weight: 700;
      color: #1e2945;
    }

    .spin-btn {
      background: #3b82f6;
      color: #ffffff;
      border: none;
      padding: 6px 16px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      box-shadow: 0 2px 6px rgba(59, 130, 246, 0.3);
      transition: background 0.2s, opacity 0.2s;
    }

    .spin-btn:active {
      opacity: 0.85;
    }

    .live-tally {
      font-size: 12px;
      color: #6b7280;
      text-align: center;
    }
  </style>
</head>
<body>

  <div class="main-grid">
    
    <div class="spinner-area">
      <div class="spinner-container">
        <div class="wheel">
          <span class="sector-label label-red">$2</span>
          <span class="sector-label label-blue">$0</span>
        </div>
        <div class="needle" id="needle"></div>
        <div class="pin"></div>
      </div>
      <button class="spin-btn" id="spinBtn" onclick="spin()">▶ Spin</button>
      <div class="live-tally" id="tally">Avg: $0.00 (0 spins)</div>
    </div>

    
    <div class="calc-card">
      <div class="calc-row row-red">
        <span><span class="badge badge-red"></span><span id="f-red"></span></span>
        <strong id="v-red"></strong>
      </div>
      <div class="calc-row row-blue">
        <span><span class="badge badge-blue"></span><span id="f-blue"></span></span>
        <strong id="v-blue"></strong>
      </div>
      <div class="calc-total">
        <span>Expected</span>
        <span id="f-total"></span>
      </div>
    </div>
  </div>

  <script>
    // Render KaTeX formulas
    katex.render("\\frac{1}{2} \\times \\$2", document.getElementById('f-red'));
    katex.render("= \\$1.00", document.getElementById('v-red'));

    katex.render("\\frac{1}{2} \\times \\$0", document.getElementById('f-blue'));
    katex.render("= \\$0.00", document.getElementById('v-blue'));

    katex.render("= \\$1.00", document.getElementById('f-total'));

    let totalSpins = 0;
    let totalWon = 0;
    let currentAngle = 0;
    let isSpinning = false;

    function spin() {
      if (isSpinning) return;
      isSpinning = true;

      // Pick red (0 to 180 deg) or blue (180 to 360 deg)
      const isRed = Math.random() < 0.5;
      const targetBase = isRed ? 20 + Math.random() * 140 : 200 + Math.random() * 140;
      const fullRotations = 360 * 3;
      currentAngle += fullRotations + (targetBase - (currentAngle % 360));
      
      const needle = document.getElementById('needle');
      needle.style.transform = `translate(-50%, -90%) rotate(${currentAngle}deg)`;

      setTimeout(() => {
        totalSpins++;
        totalWon += (isRed ? 2 : 0);
        const avg = (totalWon / totalSpins).toFixed(2);
        document.getElementById('tally').textContent = `Avg: $${avg} (${totalSpins} spins)`;
        isSpinning = false;
      }, 800);
    }
  </script>
</body>
</html>

Out of 2 spins, you can expect to land on red once ($2) and blue once ($0). You win $2 total across 2 spins, which averages out to exactly $1 per spin.

Under Scottish Curriculum for Excellence (CfE) benchmark MNU 4-22a, **Expected Frequency** is calculated as $\text{Expected Frequency} = n \cdot P(A)$, which connects directly to determining expected total winnings across $n$ independent trials.

The expected net gain/loss (expected profit) accounts for the cost of playing ($C$): $\text{Expected Net Value} = E(X) - C = \sum [(x_i - C) \cdot P(X = x_i)]$.

When the average prize equals the ticket price, we call it a **fair game**.

A game of chance is defined mathematically as 'fair' if the expected net financial value is exactly zero ($\text{Expected Net Value} = £0.00$), meaning neither the player nor the organiser has a long-term financial advantage.

What happens when the wheel changes to have more losing slices?

If $\text{Expected Net Value} < £0.00$, the game is financially unfavorable to the player and guarantees a long-term profit for the game organiser/host (typical of carnival games, commercial raffles, and national lotteries).

If $\text{Expected Net Value} > £0.00$, the player has a positive mathematical expectation over many repetitions, though short-term variance remains high.

Most real carnival games are not fair. Let's see what happens if the spinner now has 4 equal parts, but only 1 part wins $2.

<!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>Expected Value & Fairness</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
    }

    body {
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
      min-height: 100%;
    }

    .main-grid {
      display: flex;
      flex-direction: column;
      gap: 12px;
      width: 100%;
      max-width: 440px;
    }

    /* Visual 4-part wheel / grid */
    .outcomes-grid {
      display: grid;
      grid-template-columns: repeat(4, 1fr);
      gap: 6px;
      background: #f1f5f9;
      padding: 8px;
      border-radius: 12px;
    }

    .outcome-tile {
      background: #ffffff;
      border-radius: 8px;
      padding: 10px 4px;
      text-align: center;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
      transition: background-color 0.2s ease, transform 0.2s ease;
    }

    .outcome-tile.win {
      background: #ecfdf5;
      border: 1px solid #a7f3d0;
    }

    .outcome-tile.lose {
      background: #f8fafc;
      border: 1px dashed #cbd5e1;
    }

    .prob-tag {
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
    }

    .prize-val {
      font-size: 16px;
      font-weight: 700;
      color: #10b981;
    }

    .outcome-tile.lose .prize-val {
      color: #94a3b8;
    }

    /* Equation Card */
    .calc-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 12px 16px;
      box-shadow: 0 1px 4px rgba(0,0,0,0.06);
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .math-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      font-size: 15px;
      font-weight: 600;
    }

    .math-item {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 2px;
    }

    .math-label {
      font-size: 11px;
      font-weight: 500;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    .math-val {
      font-size: 17px;
      font-weight: 700;
    }

    .operator {
      font-size: 18px;
      color: #94a3b8;
      font-weight: 400;
    }

    /* Status Indicator */
    .status-badge {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      padding: 10px;
      border-radius: 10px;
      font-size: 15px;
      font-weight: 700;
      text-align: center;
      transition: background-color 0.25s ease, color 0.25s ease;
    }

    .status-negative {
      background: #fef2f2;
      color: #dc2626;
      border: 1px solid #fecaca;
    }

    .status-fair {
      background: #f0fdf4;
      color: #16a34a;
      border: 1px solid #bbf7d0;
    }

    .status-positive {
      background: #eff6ff;
      color: #2563eb;
      border: 1px solid #bfdbfe;
    }

    /* Controls */
    .controls-row {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
    }

    .ctrl-box {
      background: #ffffff;
      padding: 10px 12px;
      border-radius: 10px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      display: flex;
      flex-direction: column;
      gap: 6px;
    }

    .ctrl-label {
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      display: flex;
      justify-content: space-between;
    }

    .stepper {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 6px;
    }

    .step-btn {
      background: #f1f5f9;
      border: none;
      width: 28px;
      height: 28px;
      border-radius: 6px;
      font-size: 16px;
      font-weight: 700;
      color: #334155;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: background-color 0.15s ease;
    }

    .step-btn:active {
      background: #e2e8f0;
    }

    .val-display {
      font-size: 15px;
      font-weight: 700;
      color: #0f172a;
    }
  </style>
</head>
<body>

  <div class="main-grid">
    
    <div class="outcomes-grid">
      <div class="outcome-tile win">
        <span class="prob-tag">1 / 4</span>
        <span class="prize-val" id="tileWinVal">+£2.00</span>
      </div>
      <div class="outcome-tile lose">
        <span class="prob-tag">1 / 4</span>
        <span class="prize-val">£0.00</span>
      </div>
      <div class="outcome-tile lose">
        <span class="prob-tag">1 / 4</span>
        <span class="prize-val">£0.00</span>
      </div>
      <div class="outcome-tile lose">
        <span class="prob-tag">1 / 4</span>
        <span class="prize-val">£0.00</span>
      </div>
    </div>

    
    <div class="calc-card">
      <div class="math-row">
        <div class="math-item">
          <span class="math-label">Expected Win</span>
          <span class="math-val" id="expWin">£0.50</span>
        </div>
        <span class="operator">−</span>
        <div class="math-item">
          <span class="math-label">Cost / Play</span>
          <span class="math-val" id="costVal">£1.00</span>
        </div>
        <span class="operator">=</span>
        <div class="math-item">
          <span class="math-label">Expected Net</span>
          <span class="math-val" id="netVal">−£0.50</span>
        </div>
      </div>
    </div>

    
    <div class="status-badge status-negative" id="statusPill">
      <span id="statusIcon">✕</span>
      <span id="statusText">Unfair (−£0.50 / play)</span>
    </div>

    
    <div class="controls-row">
      <div class="ctrl-box">
        <div class="ctrl-label">
          <span>Cost</span>
        </div>
        <div class="stepper">
          <button class="step-btn" id="costDown">−</button>
          <span class="val-display" id="costDisplay">£1.00</span>
          <button class="step-btn" id="costUp">+</button>
        </div>
      </div>

      <div class="ctrl-box">
        <div class="ctrl-label">
          <span>Prize</span>
        </div>
        <div class="stepper">
          <button class="step-btn" id="prizeDown">−</button>
          <span class="val-display" id="prizeDisplay">£2.00</span>
          <button class="step-btn" id="prizeUp">+</button>
        </div>
      </div>
    </div>
  </div>

  <script>
    let cost = 1.00;
    let prize = 2.00;

    const fmt = (n) => '£' + Math.abs(n).toFixed(2);

    function update() {
      const expWinVal = prize * 0.25;
      const net = expWinVal - cost;

      // Update UI elements
      document.getElementById('costDisplay').textContent = fmt(cost);
      document.getElementById('prizeDisplay').textContent = fmt(prize);
      document.getElementById('tileWinVal').textContent = '+' + fmt(prize);

      document.getElementById('expWin').textContent = fmt(expWinVal);
      document.getElementById('costVal').textContent = fmt(cost);

      const netEl = document.getElementById('netVal');
      if (Math.abs(net) < 0.001) {
        netEl.textContent = '£0.00';
      } else if (net > 0) {
        netEl.textContent = '+' + fmt(net);
      } else {
        netEl.textContent = '−' + fmt(net);
      }

      // Update Pill status
      const pill = document.getElementById('statusPill');
      const icon = document.getElementById('statusIcon');
      const txt = document.getElementById('statusText');

      pill.className = 'status-badge';

      if (Math.abs(net) < 0.001) {
        pill.classList.add('status-fair');
        icon.textContent = '✦';
        txt.textContent = 'Fair Game (Net £0.00)';
      } else if (net > 0) {
        pill.classList.add('status-positive');
        icon.textContent = '✓';
        txt.textContent = 'Player Edge (+' + fmt(net) + ')';
      } else {
        pill.classList.add('status-negative');
        icon.textContent = '✕';
        txt.textContent = 'Unfair (−' + fmt(net) + ' / play)';
      }
    }

    document.getElementById('costDown').addEventListener('click', () => {
      if (cost > 0.25) { cost -= 0.25; update(); }
    });
    document.getElementById('costUp').addEventListener('click', () => {
      if (cost < 4.00) { cost += 0.25; update(); }
    });
    document.getElementById('prizeDown').addEventListener('click', () => {
      if (prize > 0.50) { prize -= 0.50; update(); }
    });
    document.getElementById('prizeUp').addEventListener('click', () => {
      if (prize < 8.00) { prize += 0.50; update(); }
    });

    update();
  </script>
</body>
</html>

If you buy 4 spins for $4, you only expect to win once and take home $2. You lose $2 overall, which means each spin costs you 50 cents on average.

How do raffle tickets and big lotteries compare to spinner games?

A **raffle** sells numbered tickets, mixes them in a bucket, and pulls out one winning ticket for a prize.

<!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; font-family: system-ui, -apple-system, sans-serif; }
  body { background: #f8f9fa; color: #1e2945; display: flex; flex-direction: column; align-items: center; justify-content: center; min-height: 100%; padding: 12px; }

  .wrapper { width: 100%; max-width: 480px; display: flex; flex-direction: column; gap: 12px; }

  /* Controls */
  .control-row { display: flex; align-items: center; justify-content: space-between; background: #ffffff; padding: 8px 12px; border-radius: 12px; box-shadow: 0 1px 3px rgba(0,0,0,0.06); }
  .control-label { font-size: 13px; font-weight: 600; color: #6b7280; }
  .btn-group { display: flex; gap: 6px; }
  .btn-prize {
    border: 1px solid #e5e7eb; background: #ffffff; color: #1e2945;
    padding: 6px 14px; border-radius: 8px; font-size: 14px; font-weight: 700;
    cursor: pointer; transition: all 0.2s ease;
  }
  .btn-prize.active {
    background: #f59e0b; color: #ffffff; border-color: #f59e0b;
    box-shadow: 0 2px 6px rgba(245, 158, 11, 0.35);
  }

  /* Tickets Grid */
  .tickets-box {
    background: #ffffff; padding: 12px; border-radius: 12px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.06); display: flex; flex-direction: column; gap: 8px;
  }
  .tickets-header { display: flex; justify-content: space-between; font-size: 12px; font-weight: 600; color: #6b7280; }
  .tickets-grid { display: grid; grid-template-columns: repeat(5, 1fr); gap: 6px; }
  .ticket {
    aspect-ratio: 1.4; border-radius: 6px; border: 1.5px dashed #d1d5db;
    display: flex; flex-direction: column; align-items: center; justify-content: center;
    font-size: 12px; font-weight: 700; color: #9ca3af; background: #f9fafb;
    transition: all 0.2s ease;
  }
  .ticket.win {
    border-style: solid; border-color: #f59e0b; background: #fffbeb; color: #b45309;
  }
  .ticket .icon { font-size: 13px; }

  /* Math Card */
  .math-card {
    background: #ffffff; padding: 14px; border-radius: 12px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.06); display: flex; flex-direction: column; gap: 10px;
  }
  .calc-row { display: flex; align-items: center; justify-content: space-between; font-size: 14px; }
  .calc-label { color: #6b7280; font-weight: 600; }
  .calc-value { font-weight: 800; font-size: 16px; color: #1e2945; }

  .divider { height: 1px; background: #f3f4f6; }

  /* Verdict Badge */
  .status-badge {
    padding: 8px; border-radius: 8px; text-align: center;
    font-weight: 800; font-size: 14px; letter-spacing: 0.3px;
    transition: all 0.2s ease;
  }
  .status-unfair { background: #fef2f2; color: #dc2626; }
  .status-fair { background: #fef3c7; color: #b45309; }
  .status-favorable { background: #ecfdf5; color: #059669; }
</style>
</head>
<body>

<div class="wrapper">
  
  <div class="control-row">
    <span class="control-label">Prize:</span>
    <div class="btn-group">
      <button class="btn-prize" onclick="setPrize(5)">$5</button>
      <button class="btn-prize active" onclick="setPrize(10)">$10</button>
      <button class="btn-prize" onclick="setPrize(15)">$15</button>
    </div>
  </div>

  
  <div class="tickets-box">
    <div class="tickets-header">
      <span>10 Tickets ($1 each)</span>
      <span id="winProb">P(Win) = 1/10</span>
    </div>
    <div class="tickets-grid">
      <div class="ticket win"><span class="icon">🏆</span><span id="winPrizeTag">$10</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
      <div class="ticket"><span>$0</span></div>
    </div>
  </div>

  
  <div class="math-card">
    <div class="calc-row">
      <span class="calc-label">Expected Return</span>
      <span class="calc-value" id="evVal">1/10 × $10 = $1.00</span>
    </div>
    <div class="calc-row">
      <span class="calc-label">Net per Ticket</span>
      <span class="calc-value" id="netVal">$1.00 − $1.00 = $0.00</span>
    </div>
    <div class="divider"></div>
    <div id="statusBadge" class="status-badge status-fair">Fair Game (Net = $0.00)</div>
  </div>
</div>

<script>
  function setPrize(val) {
    document.querySelectorAll('.btn-prize').forEach(b => {
      b.classList.toggle('active', b.textContent === '$' + val);
    });

    document.getElementById('winPrizeTag').textContent = '$' + val;

    const ev = val / 10;
    const net = ev - 1.0;

    document.getElementById('evVal').textContent = `1/10 × $${val} = $${ev.toFixed(2)}`;
    
    const sign = net > 0 ? '+' : '';
    document.getElementById('netVal').textContent = `$${ev.toFixed(2)} − $1.00 = ${sign}$${net.toFixed(2)}`;

    const badge = document.getElementById('statusBadge');
    badge.className = 'status-badge';
    
    if (net === 0) {
      badge.classList.add('status-fair');
      badge.textContent = 'Fair Game (EV = $1.00)';
    } else if (net < 0) {
      badge.classList.add('status-unfair');
      badge.textContent = `Unfair Game (${sign}$${net.toFixed(2)})`;
    } else {
      badge.classList.add('status-favorable');
      badge.textContent = `Favorable (+${sign}$${net.toFixed(2)})`;
    }
  }

  setPrize(10);
</script>

</body>
</html>

If 10 tickets are sold for $1 each, the organizer collects $10. If the prize is worth less than $10, the expected value of your ticket is less than what you paid.

Next time you see a game at a fair, ask yourself: if everyone played together, does the game give back as much as it takes in?

math

Calculating Expected Financial Value and Fairness in Games. Compute expected monetary value to evaluate carnival games, raffles, and lotteries.

Expected Value of Discrete Random Variables and Decision Making

Odds, Evens, and Fair Games: Assessing Risk and Expected Returns

CfE Fourth Level Mathematics: Probability, Expected Frequency, and Risk

Although the Curriculum for Excellence progression includes early intuitive ideas of chance and fairness at Early/First Level (P1–P3) through concrete games and sharing, formal computation of Expected Monetary Value ($EV$) and risk analysis is benchmarked under CfE Fourth Level (MNU 4-22a). When introducing this topic to bridge conceptual foundations to mathematical calculations, start with visual frequency trees and spinner models before transitioning to the algebraic formula $E(X) = \sum [x_i \cdot P(x_i)]$. Real-world contexts such as school fetes, raffles, and game shows provide relevant engagement for risk assessment and consumer numeracy.

Calculating Expected Financial Value and Fairness in Games

What is Expected Value?