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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<p class="byline">Andy Eisenberg</p>
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<div class="toc-title-box"><a href="notes-01-13.html" class="internal"><span class="title">Tuesday, Jan 13</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Sets" class="internal"><span class="title">Sec 1.1: Sets</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-13.html#subsec-Probability" class="internal"><span class="title">Sec 1.2: Probability</span></a></div></li>
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</ul>
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<div class="toc-title-box"><a href="notes-01-15.html" class="internal"><span class="title">Thursday, Jan 15</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Conditional-Probability" class="internal"><span class="title">Sec 1.3: Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Diagnostic-Testing" class="internal"><span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-15.html#subsec-Independent-Events" class="internal"><span class="title">Sec 1.4: Independent Events</span></a></div></li>
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</ul>
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<div class="toc-title-box"><a href="notes-01-20.html" class="internal"><span class="title">Tuesday, Jan 20</span></a></div>
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<ul class="structural toc-item-list active">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-hw1-review" class="internal"><span class="title">HW 1 Q5</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-20.html#subsec-Discrete-Random-Variables" class="internal"><span class="title">Sec 2.1: Random Variables</span></a></div></li>
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</ul>
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<div class="toc-title-box"><a href="notes-01-22.html" class="internal"><span class="title">Thursday, Jan 22</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-discrete-distributions" class="internal"><span class="title">Discrete Distributions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-22.html#subsec-continuous-distributions" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
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<div class="toc-title-box"><a href="notes-01-27.html" class="internal"><span class="title">Thursday, Jan 27</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-continuous-2" class="internal"><span class="title">Continuous Distributions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="notes-01-27.html#subsec-Joint-Distributions" class="internal"><span class="title">Joint Distributions</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-01-20"><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"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Tuesday, Jan 20</span>
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</h2>
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<section class="introduction" id="notes-01-20-2"><div class="para" id="notes-01-20-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-01-20-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></section><section class="subsection" id="subsec-hw1-review"><h3 class="heading hide-type">
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<span class="type">Subsection</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">HW 1 Q5</span>
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</h3>
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<div class="para logical" id="subsec-hw1-review-2">
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<div class="para">Write <span class="process-math">\(F\)</span> for the event that there’s a fire and <span class="process-math">\(S\)</span> for the event that there’s visible smoke. Then the information we’re given can be interpreted as:</div>
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<div class="displaymath process-math" id="subsec-hw1-review-2-3">
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\begin{align*}
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\Pr(F) \amp = 0.01 \\
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\Pr(S) \amp = 0.1 \\
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\Pr(S\mid F) \amp = 0.9
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\end{align*}
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</div>
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<div class="para">In this case, we can use the simpler version of Bayes’ Theorem:</div>
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<div class="displaymath process-math" id="subsec-hw1-review-2-4">
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\begin{gather*}
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\Pr(F \mid S) = \frac{\Pr(S \mid F)\Pr(F)}{\Pr(S)} = \dotsb
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\end{gather*}
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</div>
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<div class="para">Unlike our usual diagnostic testing examples, we do have access to the denominator probability here.</div>
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<div class="autopermalink" data-description="Paragraph"><a href="#subsec-hw1-review-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="Subsection: HW 1 Q5"><a href="#subsec-hw1-review" title="Copy heading and permalink for Subsection: HW 1 Q5" aria-label="Copy heading and permalink for Subsection: HW 1 Q5">🔗</a></div></section><section class="subsection" id="subsec-Discrete-Random-Variables"><h3 class="heading hide-type">
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<span class="type">Subsection</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">Sec 2.1: Random Variables</span>
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</h3>
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<article class="definition definition-like" id="def-RV"><h4 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">34</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para" id="def-RV-1-1">A <dfn class="terminology">random variable</dfn> is a function <span class="process-math">\(X \colon \Omega \to \R\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-RV-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 34"><a href="#def-RV" title="Copy heading and permalink for Definition 34" aria-label="Copy heading and permalink for Definition 34">🔗</a></div></article><div class="para" id="subsec-Discrete-Random-Variables-3">The idea is that <span class="process-math">\(X\)</span> is a variable representing a real number value which depends on the outcome of an experiment.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-3" 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="subsec-Discrete-Random-Variables-4"><h4 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">35</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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</h4>
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<div class="para logical" id="subsec-Discrete-Random-Variables-4-1-1">
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||
<div class="para">An experiment consists of planting 50 seeds in a garden, then growing them for 3 months. Let <span class="process-math">\(H_i\)</span> be the height of plant <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(D\)</span> be the number of seeds that didn’t sprout. Let</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-4-1-1-4">
|
||
\begin{align*}
|
||
A \amp = \text{avg height of all 50 plants} \\
|
||
\amp = \frac{H_1 + H_2 + \dotsb + H_{50}}{50}
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-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="Example 35"><a href="#subsec-Discrete-Random-Variables-4" title="Copy heading and permalink for Example 35" aria-label="Copy heading and permalink for Example 35">🔗</a></div></article><div class="para" id="subsec-Discrete-Random-Variables-5">A random variable has its own probability distribution.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="example example-like" id="subsec-Discrete-Random-Variables-6"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">36</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para" id="subsec-Discrete-Random-Variables-6-1-1">Roll a fair D6 twice. Let <span class="process-math">\(S\)</span> be the sum of the rolls. Then <span class="process-math">\(\Omega = \{(1, 1), (1, 2), \dotsc, (6, 6)\}\)</span> has 36 elements. Since the die is fair, the distribution on <span class="process-math">\(\Omega\)</span> is uniform, i.e., <span class="process-math">\(\Pr(\omega) = \frac{1}{36}\)</span> for any <span class="process-math">\(\omega \in \Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="subsec-Discrete-Random-Variables-6-1-2">
|
||
<span class="process-math">\(S\)</span> takes on the values <span class="process-math">\(2, 3, 4, \dotsc, 12\text{,}\)</span> with probabilities:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="subsec-Discrete-Random-Variables-6-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">37<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 <span class="process-math">\(S\)</span><div class="autopermalink" data-description="Table 37: Distribution for S"><a href="#subsec-Discrete-Random-Variables-6-1-3" title="Copy heading and permalink for Table 37: Distribution for S" aria-label="Copy heading and permalink for Table 37: Distribution for S">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\Pr(S = x)\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines">1/36</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines">2/36</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines">3/36</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">7</td>
|
||
<td class="c m b0 r0 l0 t0 lines">6/36</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">8</td>
|
||
<td class="c m b0 r0 l0 t0 lines">5/36</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">12</td>
|
||
<td class="c m b0 r0 l0 t0 lines">1/36</td>
|
||
</tr>
|
||
</table></div></figure> <div class="para" id="subsec-Discrete-Random-Variables-6-1-4">Note that the distribution on <span class="process-math">\(\Omega\)</span> is uniform, but the distribution on <span class="process-math">\(S\)</span> is not.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-6-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 36"><a href="#subsec-Discrete-Random-Variables-6" title="Copy heading and permalink for Example 36" aria-label="Copy heading and permalink for Example 36">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-Random-Variables-7"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">38</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="subsec-Discrete-Random-Variables-7-1-1">
|
||
<div class="para">Let <span class="process-math">\(\Omega\)</span> be a sample space and <span class="process-math">\(A \subset \Omega\)</span> an event. Let</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-7-1-1-3">
|
||
\begin{align*}
|
||
X = \begin{cases} 1 \amp x \in A \\ 0 \amp x \notin A \end{cases}
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">
|
||
<span class="process-math">\(X\)</span> is called an <dfn class="terminology">indicator random variable</dfn>, and we say "<span class="process-math">\(X\)</span> <dfn class="terminology">indicates</dfn> <span class="process-math">\(A\)</span>".</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para logical" id="subsec-Discrete-Random-Variables-7-1-2">
|
||
<div class="para">The distribution on <span class="process-math">\(X\)</span> is:</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-7-1-2-2">
|
||
\begin{align*}
|
||
\Pr(X = 1) \amp = \Pr(\{ x \mid x \in A\}) = \Pr(A) \\
|
||
\Pr(X = 0) \amp = 1 - \Pr(A)
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-7-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 38"><a href="#subsec-Discrete-Random-Variables-7" title="Copy heading and permalink for Example 38" aria-label="Copy heading and permalink for Example 38">🔗</a></div></article><article class="example example-like" id="subsec-Discrete-Random-Variables-8"><h4 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">39</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h4>
|
||
<div class="para logical" id="subsec-Discrete-Random-Variables-8-1-1">
|
||
<div class="para">Suppose we flip a coin <span class="process-math">\(n\)</span> times. Let <span class="process-math">\(H_i\)</span> indicate heads on flip <span class="process-math">\(i\text{.}\)</span> Let <span class="process-math">\(S\)</span> be the total number of heads in all flips. Then:</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-1-5">
|
||
\begin{gather*}
|
||
S = H_1 + H_2 + \dotsb + H_{n}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">If <span class="process-math">\(p\)</span> is the probability of the coin coming up heads on a flip, then <span class="process-math">\(S\)</span> has the <dfn class="terminology">binomial distribution</dfn> with parameters <span class="process-math">\(n, p\text{.}\)</span> We’ll use the notation <span class="process-math">\(S \sim \Bin(n, p)\)</span> and:</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-1-11">
|
||
\begin{gather*}
|
||
b(k) = b(k; n, p) = \Pr(S = k).
|
||
\end{gather*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="subsec-Discrete-Random-Variables-8-1-2">Suppose the coin has <span class="process-math">\(p = 0.3\)</span> and we flip it <span class="process-math">\(n = 4\)</span> times. Find <span class="process-math">\(b(2) = b(2; 4, 0.3)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para logical" id="subsec-Discrete-Random-Variables-8-1-3">
|
||
<div class="para">The relevant flip sequences are:</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-3-1">
|
||
\begin{gather*}
|
||
THHT, HHTT, TTHH, THTH, HTHT, HTTH
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Each individual sequence has a probability of <span class="process-math">\((0.3)(0.3)(0.7)(0.7) = 0.0441\text{.}\)</span> So the total probability is:</div>
|
||
<div class="displaymath process-math" id="subsec-Discrete-Random-Variables-8-1-3-3">
|
||
\begin{gather*}
|
||
b(2; 4, 0.3) = (6)(0.0441) = 0.2646.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">That is, (number of flip sequences)(probability of each sequence).</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Discrete-Random-Variables-8-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 39"><a href="#subsec-Discrete-Random-Variables-8" title="Copy heading and permalink for Example 39" aria-label="Copy heading and permalink for Example 39">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 2.1: Random Variables"><a href="#subsec-Discrete-Random-Variables" title="Copy heading and permalink for Subsection: Sec 2.1: Random Variables" aria-label="Copy heading and permalink for Subsection: Sec 2.1: Random Variables">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Jan 20"><a href="#notes-01-20" title="Copy heading and permalink for Section: Tuesday, Jan 20" aria-label="Copy heading and permalink for Section: Tuesday, Jan 20">🔗</a></div></section></div>
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