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82 lines
2.8 KiB
Markdown
82 lines
2.8 KiB
Markdown
+++
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title = "Basic probability"
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description = "Calculate theoretical and experimental probabilities for simple events and complements."
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template = "chapter.html"
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weight = 32
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[extra]
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subject = "math"
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chapter_id = "basic-probability"
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strand = "Data Handling"
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+++
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<div class="book-prose">
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<p>
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Probability measures how likely an event is, from 0 (impossible) to 1 (certain).
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</p>
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<h2 id="fundamentals" class="book-prose__heading">Fundamentals</h2>
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<p class="book-formula">\[
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P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}.
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\]</p>
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<p>
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Also: \(0\le P(E)\le1\).
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</p>
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<h2 id="complement" class="book-prose__heading">Complement rule</h2>
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<p class="book-formula">\[
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P(E')=1-P(E).
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\]</p>
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<h2 id="multiple-events" class="book-prose__heading">Combined events</h2>
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<ul>
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<li>\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)</li>
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<li>If independent: \(P(A\cap B)=P(A)P(B)\)</li>
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</ul>
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<h2 id="experimental" class="book-prose__heading">Experimental probability</h2>
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<p class="book-formula">\[
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P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}.
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\]</p>
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<h2 id="sample-space" class="book-prose__heading">Sample spaces and trees</h2>
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<p>
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Use sample-space tables or tree diagrams for multi-step outcomes.
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Multiply along branches, add across valid branches.
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</p>
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<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
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<ul>
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<li>Define event clearly before calculating.</li>
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<li>Use fractions first; round at end if needed.</li>
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<li>Check complement and total probability sanity.</li>
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<li>Use diagrams to avoid missing outcomes.</li>
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</ul>
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</div>
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<div class="book-prose">
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<details class="book-question">
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<summary class="book-question__prompt">A fair die is rolled. Find \(P(\text{prime})\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\frac{3}{6}=\frac12\) (2,3,5).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A bag has 4 red and 6 blue balls. Find \(P(\text{not red})\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\frac{6}{10}=\frac35\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">For independent events, \(P(A)=0.3,P(B)=0.4\). Find \(P(A\cap B)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(0.12\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">An event occurs 18 times in 50 trials. Estimate probability.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\frac{18}{50}=0.36\).</p>
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</div>
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</details>
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</div>
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