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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>
<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="notes-01-13"><h2 class="heading hide-type">
<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 13</span>
</h2>
<section class="introduction" id="notes-01-13-2"><div class="para" id="notes-01-13-2-1">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-01-13-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="subsec-Sets"><h3 class="heading hide-type">
<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 1.1: Sets</span>
</h3>
<div class="para" id="subsec-Sets-2">In prior math courses, you mostly asked <dfn class="terminology">deterministic</dfn> questions. Now, we need new tools to model <dfn class="terminology">randomness</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-2" 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-Sets-3"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Sets-3-1-1">A toxin molecule in a cell has a certain chance each minute to leave the cell.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-3-1-1" 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"><a href="#subsec-Sets-3" title="Copy heading and permalink for Example 1" aria-label="Copy heading and permalink for Example 1">🔗</a></div></article><article class="example example-like" id="subsec-Sets-4"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Sets-4-1-1">A patient takes a diagnostic test for a disease and wants to know the chance that they have the disease based on the test result.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-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 2"><a href="#subsec-Sets-4" title="Copy heading and permalink for Example 2" aria-label="Copy heading and permalink for Example 2">🔗</a></div></article><article class="definition definition-like" id="def-sample-space"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="def-sample-space-1-1">The <dfn class="terminology">sample space</dfn>, often denoted <span class="process-math">\(\Omega\text{,}\)</span> is the set of all possible results of an experiment. A single result is called an <dfn class="terminology">outcome</dfn>, while a collection of results is called an <dfn class="terminology">event</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-sample-space-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="Definition 3"><a href="#def-sample-space" title="Copy heading and permalink for Definition 3" aria-label="Copy heading and permalink for Definition 3">🔗</a></div></article><article class="example example-like" id="example-sample-space"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="example-sample-space-1-1">An experiment consists of rolling a standard 6-sided die (D6). The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> 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.<div class="autopermalink" data-description="Paragraph"><a href="#example-sample-space-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 4"><a href="#example-sample-space" title="Copy heading and permalink for Example 4" aria-label="Copy heading and permalink for Example 4">🔗</a></div></article><div class="para" id="subsec-Sets-7">Note: we’ll use notation like D6 to indicate a 6-sided die with faces 1, 2, 3, 4, 5, 6. Similarly, for example, D4 will indicate a 4-sided die with faces 1, 2, 3, 4.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="definition definition-like" id="def-subset"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="def-subset-1-1">The symbol <span class="process-math">\(\in\)</span> means "is an <dfn class="terminology">element</dfn> of", as in <span class="process-math">\(4 \in A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="def-subset-1-2">The symbol <span class="process-math">\(\subset\)</span> means "is a <dfn class="terminology">subset</dfn> of", as in <span class="process-math">\(A \subset \Omega\text{.}\)</span> This means that every element of the set <span class="process-math">\(A\)</span> is also an element of the set <span class="process-math">\(\Omega\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-subset-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="Definition 5"><a href="#def-subset" title="Copy heading and permalink for Definition 5" aria-label="Copy heading and permalink for Definition 5">🔗</a></div></article><article class="definition definition-like" id="def-set-operations"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="def-set-operations-1-1">
<div class="para">Consider sets <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> each contained inside <span class="process-math">\(\Omega\text{.}\)</span> We can combine sets in a variety of ways:</div>
<dl class="description-list">
<dt id="def-set-operations-1-1-4-1">Union<div class="autopermalink" data-description="Item: Union"><a href="#def-set-operations-1-1-4-1" title="Copy heading and permalink for Item: Union" aria-label="Copy heading and permalink for Item: Union">🔗</a></div>
</dt>
<dd><div class="para" id="def-set-operations-1-1-4-1-2">The <dfn class="terminology">union</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cup B = \{x \mid x \in A \text{ or } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></dd>
<dt id="def-set-operations-1-1-4-2">Intersection<div class="autopermalink" data-description="Item: Intersection"><a href="#def-set-operations-1-1-4-2" title="Copy heading and permalink for Item: Intersection" aria-label="Copy heading and permalink for Item: Intersection">🔗</a></div>
</dt>
<dd><div class="para" id="def-set-operations-1-1-4-2-2">The <dfn class="terminology">intersection</dfn> of <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> is the set <span class="process-math">\(A \cap B = \{x \mid x \in A \text{ and } x \in B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></dd>
<dt id="def-set-operations-1-1-4-3">Difference<div class="autopermalink" data-description="Item: Difference"><a href="#def-set-operations-1-1-4-3" title="Copy heading and permalink for Item: Difference" aria-label="Copy heading and permalink for Item: Difference">🔗</a></div>
</dt>
<dd><div class="para" id="def-set-operations-1-1-4-3-2">The <dfn class="terminology">set difference</dfn> <span class="process-math">\(A-B\)</span> is the set <span class="process-math">\(A - B = \{x \mid x \in A \text{ and } x \notin B\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-3-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></dd>
<dt id="def-set-operations-1-1-4-4">Complement<div class="autopermalink" data-description="Item: Complement"><a href="#def-set-operations-1-1-4-4" title="Copy heading and permalink for Item: Complement" aria-label="Copy heading and permalink for Item: Complement">🔗</a></div>
</dt>
<dd><div class="para" id="def-set-operations-1-1-4-4-2">The <dfn class="terminology">complement</dfn> of <span class="process-math">\(A\)</span> is the set <span class="process-math">\(A^c = \{x \in \Omega \mid x \notin A\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></dd>
<dt id="def-set-operations-1-1-4-5">Empty Set<div class="autopermalink" data-description="Item: Empty Set"><a href="#def-set-operations-1-1-4-5" title="Copy heading and permalink for Item: Empty Set" aria-label="Copy heading and permalink for Item: Empty Set">🔗</a></div>
</dt>
<dd><div class="para" id="def-set-operations-1-1-4-5-2">The <dfn class="terminology">empty set</dfn>, usually written <span class="process-math">\(\emptyset\)</span> or <span class="process-math">\(\{\}\text{,}\)</span> is the set which contains no elements.<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-1-1-4-5-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></dd>
</dl>
<div class="autopermalink" data-description="Paragraph"><a href="#def-set-operations-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="Definition 6"><a href="#def-set-operations" title="Copy heading and permalink for Definition 6" aria-label="Copy heading and permalink for Definition 6">🔗</a></div></article><div class="para" id="subsec-Sets-10">It’s useful sometimes to draw pictures called <dfn class="terminology">Venn diagrams</dfn> representing set interactions.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Sets-10" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="sbsgroup">
<div class="sidebyside"><div class="sbsrow" style="margin-left:5%;margin-right:5%;grid-template-columns:44.4444444444444% 44.4444444444444%;column-gap:10%;">
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-union"><img src="generated/latex-image/fig-Venn-diagram-union-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-union-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe88e;</span></summary><div id="fig-Venn-diagram-union-2-description"><div class="para" id="fig-Venn-diagram-union-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A \cup B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-union-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Union<div class="autopermalink" data-description="Figure 7"><a href="#fig-Venn-diagram-union" title="Copy heading and permalink for Figure 7" aria-label="Copy heading and permalink for Figure 7">🔗</a></div></figcaption></figure></div>
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-intersection"><img src="generated/latex-image/fig-Venn-diagram-intersection-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-intersection-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe88e;</span></summary><div id="fig-Venn-diagram-intersection-2-description"><div class="para" id="fig-Venn-diagram-intersection-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A \cap B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-intersection-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Intersection<div class="autopermalink" data-description="Figure 8"><a href="#fig-Venn-diagram-intersection" title="Copy heading and permalink for Figure 8" aria-label="Copy heading and permalink for Figure 8">🔗</a></div></figcaption></figure></div>
</div></div>
<div class="sidebyside"><div class="sbsrow" style="margin-left:5%;margin-right:5%;grid-template-columns:44.4444444444444% 44.4444444444444%;column-gap:10%;">
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-difference"><img src="generated/latex-image/fig-Venn-diagram-difference-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-difference-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe88e;</span></summary><div id="fig-Venn-diagram-difference-2-description"><div class="para" id="fig-Venn-diagram-difference-2-1-1">Venn diagram showing sets <span class="process-math">\(A, B\)</span> with the region representing <span class="process-math">\(A - B\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-difference-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Difference<div class="autopermalink" data-description="Figure 9"><a href="#fig-Venn-diagram-difference" title="Copy heading and permalink for Figure 9" aria-label="Copy heading and permalink for Figure 9">🔗</a></div></figcaption></figure></div>
<div class="sbspanel sbspanel--top top" style=""><figure class="figure figure-like" id="fig-Venn-diagram-complement"><img src="generated/latex-image/fig-Venn-diagram-complement-2.svg" role="img" class="contained" alt="described in detail following the image" aria-describedby="fig-Venn-diagram-complement-2-description"><details class="image-description" aria-live="polite"><summary title="details"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe88e;</span></summary><div id="fig-Venn-diagram-complement-2-description"><div class="para" id="fig-Venn-diagram-complement-2-1-1">Venn diagram showing set <span class="process-math">\(A \subset \Omega\)</span> with the region representing <span class="process-math">\(A^c\)</span> shaded.<div class="autopermalink" data-description="Paragraph"><a href="#fig-Venn-diagram-complement-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div></details><figcaption><span class="type">Figure</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">10<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Complement<div class="autopermalink" data-description="Figure 10"><a href="#fig-Venn-diagram-complement" title="Copy heading and permalink for Figure 10" aria-label="Copy heading and permalink for Figure 10">🔗</a></div></figcaption></figure></div>
</div></div>
</div>
<div class="autopermalink" data-description="Subsection: Sec 1.1: Sets"><a href="#subsec-Sets" title="Copy heading and permalink for Subsection: Sec 1.1: Sets" aria-label="Copy heading and permalink for Subsection: Sec 1.1: Sets">🔗</a></div></section><section class="subsection" id="subsec-Probability"><h3 class="heading hide-type">
<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 1.2: Probability</span>
</h3>
<div class="para" id="subsec-Probability-2">Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-2" 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-Probability-3"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Probability-3-1-1">An experiment consists of rolling a D6. The sample space is <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> We might assign probabilities as follows:<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="subsec-Probability-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">12<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 12: Distribution for a fair die"><a href="#subsec-Probability-3-1-2" title="Copy heading and permalink for Table 12: Distribution for a fair die" aria-label="Copy heading and permalink for Table 12: 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="subsec-Probability-3-1-3">
<div class="para">Note that we don’t have to assign the same probability to each outcome. If we do, we call this distribution <dfn class="terminology">uniform</dfn>. If we have some event, such as <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event:</div>
<div class="displaymath process-math" id="subsec-Probability-3-1-3-3">
\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="para">Note that this is the same result that we would get if we counted the total number of outcomes in <span class="process-math">\(A\)</span> and divided by the total number of outcomes in <span class="process-math">\(\Omega\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 11"><a href="#subsec-Probability-3" title="Copy heading and permalink for Example 11" aria-label="Copy heading and permalink for Example 11">🔗</a></div></article><article class="definition definition-like" id="def-cardinality"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">13</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="def-cardinality-1-1">Given a finite set <span class="process-math">\(A\text{,}\)</span> the <dfn class="terminology">cardinality</dfn> of <span class="process-math">\(A\text{,}\)</span> written <span class="process-math">\(|A|\text{,}\)</span> is the number of elements in <span class="process-math">\(A\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-cardinality-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="Definition 13"><a href="#def-cardinality" title="Copy heading and permalink for Definition 13" aria-label="Copy heading and permalink for Definition 13">🔗</a></div></article><article class="fact theorem-like" id="subsec-Probability-5"><h4 class="heading">
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">14</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="subsec-Probability-5-1-1">
<div class="para">If <span class="process-math">\(\Omega\)</span> is a finite probability space with the uniform distribution and <span class="process-math">\(A \subset \Omega\)</span> is an event, then:</div>
<div class="displaymath process-math" id="subsec-Probability-5-1-1-3">
\begin{gather*}
\Pr(A) = \frac{|A|}{|\Omega|}
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Fact 14"><a href="#subsec-Probability-5" title="Copy heading and permalink for Fact 14" aria-label="Copy heading and permalink for Fact 14">🔗</a></div></article><article class="example example-like" id="subsec-Probability-6"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">15</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Probability-6-1-1">Suppose we have a weighted die that’s much more likely to come up 6 than any other outcome.<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="subsec-Probability-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">16<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 16: Distribution for a fair die"><a href="#subsec-Probability-6-1-2" title="Copy heading and permalink for Table 16: Distribution for a fair die" aria-label="Copy heading and permalink for Table 16: 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">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">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.4</td>
</tr>
</table></div></figure> <div class="para logical" id="subsec-Probability-6-1-3">
<div class="para">With this distribution, if <span class="process-math">\(A = \{2, 4, 6\}\text{,}\)</span> then:</div>
<div class="displaymath process-math" id="subsec-Probability-6-1-3-2">
\begin{gather*}
\Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-6-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 15"><a href="#subsec-Probability-6" title="Copy heading and permalink for Example 15" aria-label="Copy heading and permalink for Example 15">🔗</a></div></article><article class="example example-like" id="subsec-Probability-7"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">17</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Probability-7-1-1">Pick a number from 1 to 10. What is the probability of picking 3?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-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-Probability-7-1-2">
<div class="para">Some considerations:</div>
<ol class="decimal" id="subsec-Probability-7-1-2-1">
<li id="subsec-Probability-7-1-2-1-1">
<div class="para" id="subsec-Probability-7-1-2-1-1-1">Are we using the uniform distribution?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 1"><a href="#subsec-Probability-7-1-2-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
</li>
<li id="subsec-Probability-7-1-2-1-2">
<div class="para" id="subsec-Probability-7-1-2-1-2-1">What even is the sample space here? Do we need to pick only integers, or is <span class="process-math">\(\pi\)</span> an outcome here?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 2"><a href="#subsec-Probability-7-1-2-1-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
</li>
<li id="subsec-Probability-7-1-2-1-3">
<div class="para" id="subsec-Probability-7-1-2-1-3-1">What would "uniform" mean if the sample space was infinite?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-7-1-2-1-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 3"><a href="#subsec-Probability-7-1-2-1-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
</li>
</ol>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-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 17"><a href="#subsec-Probability-7" title="Copy heading and permalink for Example 17" aria-label="Copy heading and permalink for Example 17">🔗</a></div></article><article class="definition definition-like" id="def-probability-distribution"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">18</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<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>
<ol class="decimal" id="def-probability-distribution-1-1-3">
<li id="def-probability-distribution-1-1-3-1">
<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>
</div>
<div class="autopermalink" data-description="Item 1"><a href="#def-probability-distribution-1-1-3-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
</li>
<li id="def-probability-distribution-1-1-3-2">
<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>
</div>
<div class="autopermalink" data-description="Item 2"><a href="#def-probability-distribution-1-1-3-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
</li>
<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>
</div>
<div class="autopermalink" data-description="Item 3"><a href="#def-probability-distribution-1-1-3-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
</li>
</ol>
<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>
</div>
<div class="autopermalink" data-description="Definition 18"><a href="#def-probability-distribution" title="Copy heading and permalink for Definition 18" aria-label="Copy heading and permalink for Definition 18">🔗</a></div></article><article class="example example-like" id="subsec-Probability-9"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">19</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Probability-9-1-1">Suppose we flip a coin until we see heads. The sample space is <span class="process-math">\(\Omega = \{H, TH, TTH, TTTH, \dotsc\}\text{.}\)</span> What would "uniform" mean here? Is there any way we could assign the same probability to each individual outcome here?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="subsec-Probability-9-1-2">Instead of "uniform", what if we want to treat the coin as a fair coin? Then what would the distribution be?<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-9-1-2" 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-Probability-9-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">20<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution assuming a fair coin<div class="autopermalink" data-description="Table 20: Distribution assuming a fair coin"><a href="#subsec-Probability-9-1-3" title="Copy heading and permalink for Table 20: Distribution assuming a fair coin" aria-label="Copy heading and permalink for Table 20: Distribution assuming a fair coin">🔗</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"># flips</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"><span class="process-math">\(H\)</span></td>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/2\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TH\)</span></td>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/4\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TTH\)</span></td>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/8\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(TTTH\)</span></td>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/16\)</span></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>
<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"><span class="process-math">\(T\dotsm TH\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(n\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/2^n\)</span></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>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Example 19"><a href="#subsec-Probability-9" title="Copy heading and permalink for Example 19" aria-label="Copy heading and permalink for Example 19">🔗</a></div></article><article class="example example-like" id="subsec-Probability-10"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">21</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="subsec-Probability-10-1-1">One last (very useful!) observation. Recall: 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="#subsec-Probability-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="subsec-Probability-10-1-2">
<div class="para">For any event <span class="process-math">\(A\text{,}\)</span> <span class="process-math">\(A\cap A^c = \emptyset\)</span> and <span class="process-math">\(A \cup A^c = \Omega\text{.}\)</span> So:</div>
<div class="displaymath process-math" id="subsec-Probability-10-1-2-4">
\begin{gather*}
1 = \Pr(\underbrace{A \cup A^c}_{\Omega}) = \Pr(A) + \Pr(A^c).
\end{gather*}
</div>
<div class="para">We can rewrite this in two useful ways:</div>
<div class="displaymath process-math" id="subsec-Probability-10-1-2-5">
\begin{align*}
\Pr(A) \amp = 1 - \Pr(A^c) \\
\Pr(A^c) \amp = 1 - \Pr(A)
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#subsec-Probability-10-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 21"><a href="#subsec-Probability-10" title="Copy heading and permalink for Example 21" aria-label="Copy heading and permalink for Example 21">🔗</a></div></article><div class="autopermalink" data-description="Subsection: Sec 1.2: Probability"><a href="#subsec-Probability" title="Copy heading and permalink for Subsection: Sec 1.2: Probability" aria-label="Copy heading and permalink for Subsection: Sec 1.2: Probability">🔗</a></div></section><div class="autopermalink" data-description="Section: Tuesday, Jan 13"><a href="#notes-01-13" title="Copy heading and permalink for Section: Tuesday, Jan 13" aria-label="Copy heading and permalink for Section: Tuesday, Jan 13">🔗</a></div></section></div>
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