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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Probability"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Definition of Probability</span>
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<article class="definition definition-like" id="def-probability-distribution"><h3 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="def-probability-distribution-1-1">
<div class="para">A <dfn class="terminology">probability distribution</dfn> on a sample space <span class="process-math">\(\Omega\)</span> assigns probabilities to every event, satisfying the following conditions:</div>
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<div class="para" id="def-probability-distribution-1-1-3-1-1">
<span class="process-math">\(\Pr(\Omega) = 1\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para" id="def-probability-distribution-1-1-3-2-1">
<span class="process-math">\(0 \leq \Pr(A) \leq 1\)</span> for any event <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<li id="def-probability-distribution-1-1-3-3">
<div class="para" id="def-probability-distribution-1-1-3-3-1">If <span class="process-math">\(A \cap B = \emptyset\text{,}\)</span> then <span class="process-math">\(\Pr(A\cup B) = \Pr(A) + \Pr(B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Paragraph"><a href="#def-probability-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 1.2.1"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 1.2.1" aria-label="Copy heading and permalink for Definition 1.2.1">🔗</a></div></article><div class="para" id="sec-Probability-3">For small probability spaces (i.e., with finitely many outcomes in the sample space), we’ll usually assign probabilities to each individual outcome, and perhaps list them in a table. Then, to find the probability of any event, simply add together the probabilities of each outcome in that event.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="example example-like" id="sec-Probability-4"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="sec-Probability-4-1-1">An experiment consists of rolling a standard 6-sided die. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> The probability distribution (assuming a fair die) is shown below.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="sec-Probability-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution for a fair die<div class="autopermalink" data-description="Table 1.2.3: Distribution for a fair die"><a href="#sec-Probability-4-1-2" title="Copy heading and permalink for Table 1.2.3: Distribution for a fair die" aria-label="Copy heading and permalink for Table 1.2.3: Distribution for a fair die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/6\)</span></td>
</tr>
</table></div></figure> <div class="para logical" id="sec-Probability-4-1-3">
<div class="para">One possible event is <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> i.e., the event that the result of the roll is even. The probability of <span class="process-math">\(A\)</span> is:</div>
<div class="displaymath process-math" id="sec-Probability-4-1-3-3">
\begin{gather*}
\Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-4-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Example 1.2.2"><a href="#sec-Probability-4" title="Copy heading and permalink for Example 1.2.2" aria-label="Copy heading and permalink for Example 1.2.2">🔗</a></div></article><article class="definition definition-like" id="def-discrete-uniform-distribution"><h3 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="def-discrete-uniform-distribution-1-1">Let <span class="process-math">\(\Omega = \{x_1, x_2, \dotsc, x_n\}\)</span> be a sample space and <span class="process-math">\(A\subset \Omega\)</span> an event. We refer to the distribution in which <span class="process-math">\(\Pr(x_i) = \frac{1}{n}\)</span> for all <span class="process-math">\(i\)</span> as the <dfn class="terminology">uniform distribution</dfn>. In this case, it follows that <span class="process-math">\(\Pr(A) = \frac{|A|}{|\Omega|}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-discrete-uniform-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Definition 1.2.4"><a href="#def-discrete-uniform-distribution" title="Copy heading and permalink for Definition 1.2.4" aria-label="Copy heading and permalink for Definition 1.2.4">🔗</a></div></article><article class="example example-like" id="sec-Probability-6"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="sec-Probability-6-1-1">A person picks a random number from 1 to 10. What is the probability that they picked 3?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 1.2.5"><a href="#sec-Probability-6" title="Copy heading and permalink for Example 1.2.5" aria-label="Copy heading and permalink for Example 1.2.5">🔗</a></div></article><section class="exercises" id="exercises-Probability"><h3 class="heading hide-type">
<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>
</h3>
<article class="exercise exercise-like" id="exercises-Probability-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-1-1-1">Consider the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\)</span> with probability distribution below. Calculate the probabilities of <span class="process-math">\(A = \{1, 3, 7, 8\}\text{,}\)</span> <span class="process-math">\(B = \{2, 3, 6, 7\}\text{,}\)</span> <span class="process-math">\(A\cup B\text{,}\)</span> and <span class="process-math">\(A \cap B\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="exercises-Probability-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1.2.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 1.2.6: "><a href="#exercises-Probability-1-1-2" title="Copy heading and permalink for Table 1.2.6: " aria-label="Copy heading and permalink for Table 1.2.6: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure> <div class="autopermalink" data-description="Exercise 1.2.1"><a href="#exercises-Probability-1" title="Copy heading and permalink for Exercise 1.2.1" aria-label="Copy heading and permalink for Exercise 1.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="exercises-Probability-2-1"><div class="para" id="exercises-Probability-2-1-1">Suppose we flip a coin two times. Answer the questions below. What about three flips? What about four flips?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="exercises-Probability-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Probability-2-2-1-1">Write all outcomes in the sample space <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 1.2.2.a"><a href="#exercises-Probability-2-2" title="Copy heading and permalink for Task 1.2.2.a" aria-label="Copy heading and permalink for Task 1.2.2.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Probability-2-3-1-1">Make a probability distribution table for <span class="process-math">\(\Omega\)</span> assuming the coin is fair.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.2.2.b"><a href="#exercises-Probability-2-3" title="Copy heading and permalink for Task 1.2.2.b" aria-label="Copy heading and permalink for Task 1.2.2.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-2-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-Probability-2-4-1-1">Make a probability distribution table assuming the coin comes up heads with probability 0.3.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-2-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.2.2.c"><a href="#exercises-Probability-2-4" title="Copy heading and permalink for Task 1.2.2.c" aria-label="Copy heading and permalink for Task 1.2.2.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.2.2"><a href="#exercises-Probability-2" title="Copy heading and permalink for Exercise 1.2.2" aria-label="Copy heading and permalink for Exercise 1.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="exercises-Probability-3-1"><div class="para" id="exercises-Probability-3-1-1">Suppose we roll a die two times. Answer the questions below.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="exercises-Probability-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Probability-3-2-1-1">Write all outcomes in the sample space <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 1.2.3.a"><a href="#exercises-Probability-3-2" title="Copy heading and permalink for Task 1.2.3.a" aria-label="Copy heading and permalink for Task 1.2.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Probability-3-3-1-1">Make a probability distribution table for <span class="process-math">\(\Omega\)</span> assuming the die is fair.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.2.3.b"><a href="#exercises-Probability-3-3" title="Copy heading and permalink for Task 1.2.3.b" aria-label="Copy heading and permalink for Task 1.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Probability-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-Probability-3-4-1-1">Let <span class="process-math">\(A\)</span> be the event that the second roll is higher than the first, and let <span class="process-math">\(B\)</span> be the event that the first roll is even. Find <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> and <span class="process-math">\(\Pr(A \mid B)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 1.2.3.c"><a href="#exercises-Probability-3-4" title="Copy heading and permalink for Task 1.2.3.c" aria-label="Copy heading and permalink for Task 1.2.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 1.2.3"><a href="#exercises-Probability-3" title="Copy heading and permalink for Exercise 1.2.3" aria-label="Copy heading and permalink for Exercise 1.2.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-4-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.4"><a href="#exercises-Probability-4" title="Copy heading and permalink for Exercise 1.2.4" aria-label="Copy heading and permalink for Exercise 1.2.4">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-5-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.5"><a href="#exercises-Probability-5" title="Copy heading and permalink for Exercise 1.2.5" aria-label="Copy heading and permalink for Exercise 1.2.5">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-6-1-1">A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> find the probability of the toxin molecule leaving the cell during the <span class="process-math">\(n\)</span>th minute. What is the probability of the molecule leaving the cell during the first 3 minutes?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 1.2.6"><a href="#exercises-Probability-6" title="Copy heading and permalink for Exercise 1.2.6" aria-label="Copy heading and permalink for Exercise 1.2.6">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Probability-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Probability-7-1-1">Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> and for each value of <span class="process-math">\(0\leq k \leq n\text{,}\)</span> find the probability that exactly <span class="process-math">\(k\)</span> toxin molecules remain in the cell after the <span class="process-math">\(n\)</span>th minute.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Probability-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 1.2.7"><a href="#exercises-Probability-7" title="Copy heading and permalink for Exercise 1.2.7" aria-label="Copy heading and permalink for Exercise 1.2.7">🔗</a></div></article><div class="autopermalink" data-description="Exercises 1.2"><a href="#exercises-Probability" title="Copy heading and permalink for Exercises 1.2" aria-label="Copy heading and permalink for Exercises 1.2">🔗</a></div></section><div class="autopermalink" data-description="Section 1.2: Definition of Probability"><a href="#sec-Probability" title="Copy heading and permalink for Section 1.2: Definition of Probability" aria-label="Copy heading and permalink for Section 1.2: Definition of Probability">🔗</a></div></section></div>
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