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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
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<div class="toc-title-box"><a href="sec-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
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<ul class="structural toc-item-list active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<li class="toc-item toc-section">
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Central Limit Theorem</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div></li>
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Independent-Events"><h2 class="heading hide-type">
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<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Independent Events</span>
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</h2>
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<div class="para" id="sec-Independent-Events-2">The idea of conditional probability is that knowledge of one event can change our understanding of the probability of another. But it’s also important to understand when this is not the case.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Independent-Events-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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<article class="definition definition-like" id="def-independent-events"><h3 class="heading">
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<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h3>
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<div class="para" id="def-independent-events-1-1">Events <span class="process-math">\(A, B\)</span> are <dfn class="terminology">independent</dfn> if <span class="process-math">\(\Pr(A \mid B) = \Pr(A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-independent-events-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="autopermalink" data-description="Definition 1.4.1"><a href="#def-independent-events" title="Copy heading and permalink for Definition 1.4.1" aria-label="Copy heading and permalink for Definition 1.4.1">🔗</a></div></article><div class="para logical" id="sec-Independent-Events-4">
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<div class="para">The equation <span class="process-math">\(\Pr(A\mid B) = \Pr(A)\)</span> very concretely says: knowledge that the event <span class="process-math">\(B\)</span> has occurred does not change our understanding of <span class="process-math">\(\Pr(A)\text{.}\)</span> Assuming <span class="process-math">\(\Pr(A) \neq 0\text{,}\)</span> this is equivalent to <span class="process-math">\(\Pr(A\cap B) = \Pr(A)\Pr(B)\text{,}\)</span> since:</div>
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<div class="displaymath process-math" id="sec-Independent-Events-4-6">
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\begin{align*}
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\Pr(A\mid B) \amp = \Pr(A) \\
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\frac{\Pr(A\cap B)}{\Pr(B)} \amp = \Pr(A) \\
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\Pr(A\cap B) \amp = \Pr(A)\Pr(B)
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\end{align*}
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</div>
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<div class="para">This latter form of the equation is often more useful in calculations.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#sec-Independent-Events-4" 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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<article class="example example-like" id="example-rolls-independent"><h3 class="heading">
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<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.4.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h3>
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<div class="para" id="example-rolls-independent-1-1">In <a href="sec-Conditional-Probability.html#example-rolls-conditional" class="xref" data-knowl="./knowl/xref/example-rolls-conditional.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 1.3.2">Example 1.3.2</a>, we considered an experiment in which we rolled a fair 6-sided die twice. We defined events <span class="process-math">\(A\)</span> that the sum of the rolls is at least 10 and <span class="process-math">\(B\)</span> that the first roll is a 6, and we found that <span class="process-math">\(\Pr(A \mid B) \neq \Pr(A)\text{,}\)</span> so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <div class="para logical" id="example-rolls-independent-1-2">
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<div class="para">Now consider the event <span class="process-math">\(C\)</span> that the sum of the rolls is 7, which sounds very similar to the event <span class="process-math">\(A\text{.}\)</span> We have:</div>
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<div class="displaymath process-math" id="example-rolls-independent-1-2-3">
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\begin{align*}
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C \amp = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \\
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\Pr(C) \amp = \frac{6}{36} = \frac{1}{6} \\
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B\cap C \amp = \{(6, 1)\} \\
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\Pr(B\cap C) \amp = \frac{1}{36} \\
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\text{therefore: } \Pr(C\mid B) \amp = \frac{\Pr(C\cap B)}{\Pr(B)} = \frac{1/36}{1/6} = \frac{1}{6} = \Pr(C)
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\end{align*}
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</div>
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<div class="para">So, even though the descriptions of events <span class="process-math">\(A\)</span> and <span class="process-math">\(C\)</span> are very similar, the event <span class="process-math">\(C\)</span> is independent with <span class="process-math">\(B\text{,}\)</span> while <span class="process-math">\(A\)</span> is not.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#example-rolls-independent-1-2" 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="Example 1.4.2"><a href="#example-rolls-independent" title="Copy heading and permalink for Example 1.4.2" aria-label="Copy heading and permalink for Example 1.4.2">🔗</a></div></article><section class="exercises" id="exercises-Independent-Events"><h3 class="heading hide-type">
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<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
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</h3>
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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="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="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="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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