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@@ -335,12 +335,100 @@ B\cap C \amp = \{(6, 1)\} \\
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<article class="exercise exercise-like" id="exercises-Independent-Events-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Independent-Events-1-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Independent-Events-1-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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<div class="para logical" id="exercises-Independent-Events-1-2-1">
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<div class="displaymath process-math" id="exercises-Independent-Events-1-2-1-1">
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\begin{align*}
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A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), \\
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\amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), \\
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\amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} \\
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B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
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\amp (2, 3), (2, 4), (2, 5), (2, 6), \\
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\amp (3, 4), (3, 5), (3, 6), \\
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\amp (4, 5), (4, 6), \\
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\amp (5, 6)\} \\
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A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
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\end{align*}
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</div>
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<div class="para">So <span class="process-math">\(\Pr(A) = \frac{18}{36} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{15}{36} = \frac{5}{12}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Finally:</div>
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<div class="displaymath process-math" id="exercises-Independent-Events-1-2-1-5">
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\begin{align*}
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\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \neq \Pr(A),
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\end{align*}
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</div>
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<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Answer 1.4.1.1"><a href="#exercises-Independent-Events-1-2" title="Copy heading and permalink for Answer 1.4.1.1" aria-label="Copy heading and permalink for Answer 1.4.1.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 1.4.1"><a href="#exercises-Independent-Events-1" title="Copy heading and permalink for Exercise 1.4.1" aria-label="Copy heading and permalink for Exercise 1.4.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Independent-Events-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Independent-Events-2-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Independent-Events-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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<div class="para logical" id="exercises-Independent-Events-2-2-1">
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<div class="displaymath process-math" id="exercises-Independent-Events-2-2-1-1">
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\begin{align*}
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A \amp = \{ HHH, HHT, TTH, TTT \} \\
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B \amp = \{ HHH, HHT, HTH, THH \} \\
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A\cap B \amp = \{HHH, HHT\}
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\end{align*}
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</div>
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<div class="para">So <span class="process-math">\(\Pr(A) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}\text{.}\)</span> Finally:</div>
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<div class="displaymath process-math" id="exercises-Independent-Events-2-2-1-5">
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\begin{align*}
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\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} = \Pr(A),
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\end{align*}
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</div>
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<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Answer 1.4.2.1"><a href="#exercises-Independent-Events-2-2" title="Copy heading and permalink for Answer 1.4.2.1" aria-label="Copy heading and permalink for Answer 1.4.2.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 1.4.2"><a href="#exercises-Independent-Events-2" title="Copy heading and permalink for Exercise 1.4.2" aria-label="Copy heading and permalink for Exercise 1.4.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Independent-Events-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Independent-Events-3-1-1">Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{3, 4, 5\}\)</span> be events in the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> Create a probability distribution for <span class="process-math">\(\Omega\)</span> so that <span class="process-math">\(A, B\)</span> are independent.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Independent-Events-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
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<figure class="table table-like" id="exercises-Independent-Events-3-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" data-description="Table 1.4.3: Example Distribution"><a href="#exercises-Independent-Events-3-2-1" title="Copy heading and permalink for Table 1.4.3: Example Distribution" aria-label="Copy heading and permalink for Table 1.4.3: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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<tr>
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<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
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<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">1</td>
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<td class="c m b0 r0 l0 t0 lines">0.1</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">2</td>
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<td class="c m b0 r0 l0 t0 lines">0.2</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">3</td>
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<td class="c m b0 r0 l0 t0 lines">0.2</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">4</td>
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<td class="c m b0 r0 l0 t0 lines">0.1</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">5</td>
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<td class="c m b0 r0 l0 t0 lines">0.1</td>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">6</td>
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<td class="c m b0 r0 l0 t0 lines">0.3</td>
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</tr>
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</table></div></figure><div class="para logical" id="exercises-Independent-Events-3-2-2">
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<div class="para">Now <span class="process-math">\(\Pr(A) = 0.5\text{,}\)</span> <span class="process-math">\(\Pr(B) = 0.4\text{,}\)</span> and</div>
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<div class="displaymath process-math" id="exercises-Independent-Events-3-2-2-3">
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\begin{gather*}
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\Pr(A\cap B) = 0.2 = (0.5)(0.4) = \Pr(A)\Pr(B),
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\end{gather*}
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</div>
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<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Independent-Events-3-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Answer 1.4.3.1"><a href="#exercises-Independent-Events-3-2" title="Copy heading and permalink for Answer 1.4.3.1" aria-label="Copy heading and permalink for Answer 1.4.3.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 1.4.3"><a href="#exercises-Independent-Events-3" title="Copy heading and permalink for Exercise 1.4.3" aria-label="Copy heading and permalink for Exercise 1.4.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.4"><a href="#exercises-Independent-Events" title="Copy heading and permalink for Exercises 1.4" aria-label="Copy heading and permalink for Exercises 1.4">🔗</a></div></section><div class="autopermalink" data-description="Section 1.4: Independent Events"><a href="#sec-Independent-Events" title="Copy heading and permalink for Section 1.4: Independent Events" aria-label="Copy heading and permalink for Section 1.4: Independent Events">🔗</a></div></section></div>
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