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@@ -25,19 +25,17 @@ <h2>Overview</h2>
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<h3>Competencies</h3>
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<ul>
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<li>Identifies probability as a measure of likelihood</li>
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<li>Describes outcomes and sample spaces</li>
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<li>Estimates event frequency from real or simulated data</li>
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<li>Compares probabilities</li>
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<li>Interprets relationships between events using tables or diagrams</li>
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<li>Identify probability as a measure of likelihood</li>
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<li>Describe outcomes and sample spaces</li>
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<li>Estimate event frequency from real or simulated data</li>
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<li>Compare probabilities</li>
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<li>Interpret relationships between events using tables or diagrams</li>
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</ul>
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</section>
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<sectionid="getting-started">
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<h2>Getting Started</h2>
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<ol>
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<li><strong>Choose a mode:</strong><em>One event</em> (single sample space) or <em>Two events</em> (two-way table).</li>
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<li><strong>Pick a device</strong> (coin / die / spinner / custom) and optionally adjust its bias.</li>
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<li><strong>Define Event E</strong> by selecting outcomes (e.g., “Heads”, or “Even” by selecting 2, 4, 6).</li>
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<li><strong>Run trials</strong> using <em>Run</em>, the +1/+10/+100/+1000 buttons, or <em>Start auto</em>.</li>
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<li><strong>Interpret the results</strong> using the charts and tables (look at how the estimates stabilize as trials increase).</li>
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<p>Lists theoretical probability, count, relative frequency, and the difference (<strong>Δ</strong>) between experiment and theory for each outcome. Selected outcomes (Event E) are highlighted.</p>
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<h3>Two Events: Heatmap + Two-way Table</h3>
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<p>Switch to <em>Two events</em> to explore joint outcomes. Click a cell to see:</p>
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<p>Click a cell in the heatmap to see:</p>
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<ul>
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<li><strong>P(A ∩ B)</strong> (joint probability estimate)</li>
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<li><strong>P(A | B)</strong> and <strong>P(B | A)</strong> (conditional probability estimate)</li>
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<li>An "independence check" comparing <strong>P(A)·P(B)</strong> to <strong>P(A ∩ B)</strong></li>
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</ul>
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<h3>Seeded Runs</h3>
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<p>Use a <strong>seed</strong> to repeat the same random sequence and compare runs.</p>
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</section>
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<sectionid="workflow">
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<h2>Learning Workflow</h2>
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<p>Use these short tasks to build intuition quickly:</p>
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<ol>
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<li><strong>Sample space:</strong> Choose a device and list its outcomes (e.g., die: 1–6).</li>
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<li><strong>Event building:</strong> Define an event (e.g., “Even” on a die) by selecting outcomes.</li>
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<li><strong>Convergence:</strong> Run 10, 100, 1,000 trials. Describe how the estimate changes.</li>
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<li><strong>Comparing probabilities:</strong> Change the device bias (e.g., increase P(6)) and predict what will happen before running.</li>
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<li><strong>Relationships:</strong> In two-event mode, compare <em>Independent</em> vs <em>Dependent</em>. Explain what changes in the two-way table.</li>
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</ol>
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<h3>Tips & Best Practices</h3>
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<ul>
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<li>Small samples are noisy on purpose—don’t “fix” the randomness.</li>
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<li>If you change the device bias, treat it as a new experiment and reset.</li>
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<li>Use the tables to justify claims (don’t rely only on what “feels” likely).</li>
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</ul>
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</section>
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<sectionid="shortcuts">
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<h2>Shortcuts</h2>
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<ul>
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<li><strong>ESC</strong> - Close help modal</li>
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<li><strong>Enter</strong> (in the Seed input) - Apply seed</li>
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<li><strong>Click a heatmap/two-way cell</strong> - Show joint + conditional probabilities</li>
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</ul>
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</section>
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<summary>What should I look for in the two-way table?</summary>
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<p>Start with the joint counts. Then click a cell to compare <strong>P(A ∩ B)</strong> to <strong>P(A)·P(B)</strong>. If events are independent, these values should be close (with enough trials).</p>
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</details>
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<details>
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<summary>Is anything saved?</summary>
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<p>The lab saves basic settings (mode, device choices, bias sliders, event selection, seed) in your browser so you can continue where you left off. Trial results are not persisted.</p>
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