529 lines
16 KiB
XML
529 lines
16 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<section xml:id="notes-week-01">
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<title>Week 1</title>
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<introduction>
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<p>
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This is an outline of the topics we covered in the first week of class.
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These notes are <em>not</em> a substitute for your own note-taking.
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I highly recommend that you take your own notes during class.
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If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
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</p>
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</introduction>
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<subsection>
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<title>Tuesday 1/13</title>
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<subsubsection xml:id="subsubsec-Sets">
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<title>Sec 1.1: Sets</title>
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<p>
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In prior math courses, you mostly asked <term>deterministic</term> questions.
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Now, we need new tools to model <term>randomness</term>.
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</p>
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<example>
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<statement>
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<p>
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A toxin molecule in a cell has a certain chance each minute to leave the cell.
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</p>
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</statement>
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</example>
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<example>
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<statement>
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<p>
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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.
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</p>
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</statement>
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</example>
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<definition xml:id="def-sample-space">
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<statement>
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<p>
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The <term>sample space</term>, often denoted <m>\Omega</m>, is the set of all possible results of an experiment.
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A single result is called an <term>outcome</term>, while a collection of results is called an <term>event</term>.
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</p>
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</statement>
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</definition>
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<example xml:id="example-sample-space">
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<statement>
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<p>
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An experiment consists of rolling a standard 6-sided die (D6).
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The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
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One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the result of the roll is even.
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</p>
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</statement>
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</example>
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<p>
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Note: we'll use notation like D6 to indicate a 6-sided die with faces 1, 2, 3, 4, 5, 6.
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Similarly, for example, D4 will indicate a 4-sided die with faces 1, 2, 3, 4.
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</p>
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<definition xml:id="def-subset">
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<statement>
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<p>
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The symbol <m>\in</m> means "is an <term>element</term> of", as in <m>4 \in A</m>.
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</p>
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<p>
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The symbol <m>\subset</m> means "is a <term>subset</term> of", as in <m>A \subset \Omega</m>.
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This means that every element of the set <m>A</m> is also an element of the set <m>\Omega</m>.
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</p>
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</statement>
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</definition>
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<definition xml:id="def-set-operations">
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<statement>
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<p>
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Consider sets <m>A</m> and <m>B</m>, each contained inside <m>\Omega</m>.
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We can combine sets in a variety of ways: <dl>
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<li>
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<title>Union</title>
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<p>
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The <term>union</term> of <m>A</m> and <m>B</m> is the set <m>A \cup B = \{x \mid x \in A \text{ or } x \in B\}</m>.
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</p>
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</li>
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<li>
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<title>Intersection</title>
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<p>
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The <term>intersection</term> of <m>A</m> and <m>B</m> is the set <m>A \cap B = \{x \mid x \in A \text{ and } x \in B\}</m>.
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</p>
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</li>
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<li>
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<title>Difference</title>
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<p>
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The <term>set difference</term> <m>A-B</m> is the set <m>A - B = \{x \mid x \in A \text{ and } x \notin B\}</m>.
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</p>
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</li>
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<li>
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<title>Complement</title>
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<p>
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The <term>complement</term> of <m>A</m> is the set <m>A^c = \{x \in \Omega \mid x \notin A\}</m>.
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</p>
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</li>
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<li>
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<title>Empty Set</title>
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<p>
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The <term>empty set</term>, usually written <m>\emptyset</m> or <m>\{\}</m>, is the set which contains no elements.
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</p>
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</li>
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</dl>
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</p>
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</statement>
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</definition>
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<p>
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It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing set interactions.
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</p>
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<sbsgroup widths="40% 40%">
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<sidebyside>
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<figure xml:id="fig-Venn-diagram-union">
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<caption>Union</caption>
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<image>
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<description>
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<p>
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Venn diagram showing sets <m>A, B</m> with the region representing <m>A \cup B</m> shaded.
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(180:1.75cm) circle (2.5cm)}
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\def\secondcircle{(0:1.75cm) circle (2.5cm)}
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\fill[gray!30] \firstcircle;
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\fill[gray!30] \secondcircle;
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\draw \firstcircle node[text=black,left] {$A$};
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\draw \secondcircle node [text=black,right] {$B$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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<figure xml:id="fig-Venn-diagram-intersection">
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<caption>Intersection</caption>
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<image>
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<description>
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<p>
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Venn diagram showing sets <m>A, B</m> with the region representing <m>A \cap B</m> shaded.
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(180:1.75cm) circle (2.5cm)}
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\def\secondcircle{(0:1.75cm) circle (2.5cm)}
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\fill[white] \firstcircle;
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\fill[white] \secondcircle;
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\begin{scope}
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\clip \firstcircle;
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\clip \secondcircle;
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\fill[gray!30] \firstcircle;
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\end{scope}
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\draw \firstcircle node[text=black,left] {$A$};
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\draw \secondcircle node [text=black,right] {$B$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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</sidebyside>
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<sidebyside>
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<figure xml:id="fig-Venn-diagram-difference">
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<caption>Difference</caption>
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<image>
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<description>
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<p>
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Venn diagram showing sets <m>A, B</m> with the region representing <m>A - B</m> shaded.
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(180:1.75cm) circle (2.5cm)}
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\def\secondcircle{(0:1.75cm) circle (2.5cm)}
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\fill[gray!30] \firstcircle;
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\fill[white] \secondcircle;
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\draw \firstcircle node[text=black,left] {$A$};
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\draw \secondcircle node [text=black,right] {$B$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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<figure xml:id="fig-Venn-diagram-complement">
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<caption>Complement</caption>
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<image>
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<description>
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<p>
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Venn diagram showing set <m>A \subset \Omega</m> with the region representing <m>A^c</m> shaded.
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(0, 0) circle (2.5cm)}
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\fill [gray!30] (-4, -3) rectangle (5, 3);
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\fill[white] \firstcircle;
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\draw \firstcircle node[text=black] {$A$};
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\draw (-4, -3) rectangle (5, 3) node [text=black,right] {$\Omega$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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</sidebyside>
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</sbsgroup>
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</subsubsection>
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<subsubsection xml:id="subsubsec-Probability">
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<title>Sec 1.2: Probability</title>
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<p>
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Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.
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</p>
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<example>
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<statement>
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<p>
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An experiment consists of rolling a D6.
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The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
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We might assign probabilities as follows:
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</p>
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<table>
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<title>Distribution for a fair die</title>
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<tabular halign="center">
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<row bottom="minor">
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<cell><m>x</m></cell>
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<cell><m>\Pr(x)</m></cell>
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</row>
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<row>
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<cell>1</cell>
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<cell><m>1/6</m></cell>
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</row>
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<row>
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<cell>2</cell>
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<cell><m>1/6</m></cell>
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</row>
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<row>
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<cell>3</cell>
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<cell><m>1/6</m></cell>
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</row>
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<row>
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<cell>4</cell>
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<cell><m>1/6</m></cell>
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</row>
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<row>
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<cell>5</cell>
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<cell><m>1/6</m></cell>
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</row>
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<row>
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<cell>6</cell>
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<cell><m>1/6</m></cell>
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</row>
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</tabular>
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</table>
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<p>
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Note that we don't have to assign the same probability to each outcome.
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If we do, we call this distribution <term>uniform</term>.
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If we have some event, such as <m>A = \{2, 4, 6\}</m>, then we calculate the probability of the event by adding together the probabilities of the outcomes that make up the event:
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<md>
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<mrow> \Pr(A) = \Pr(2) + \Pr(4) + \Pr(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} </mrow>
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</md>
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Note that this is the same result that we would get if we counted the total number of outcomes in <m>A</m> and divided by the total number of outcomes in <m>\Omega</m>.
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</p>
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</statement>
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</example>
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<definition xml:id="def-cardinality">
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<statement>
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<p>
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Given a finite set <m>A</m>, the <term>cardinality</term> of <m>A</m>, written <m>|A|</m>, is the number of elements in <m>A</m>.
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</p>
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</statement>
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</definition>
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<fact>
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<statement>
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<p>
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If <m>\Omega</m> is a finite probability space with the uniform distribution and <m>A \subset \Omega</m> is an event, then:
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<md>
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<mrow> \Pr(A) = \frac{|A|}{|\Omega|} </mrow>
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</md>
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</p>
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</statement>
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</fact>
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<example>
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<statement>
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<p>
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Suppose we have a weighted die that's much more likely to come up 6 than any other outcome.
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</p>
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<table>
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<title>Distribution for a fair die</title>
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<tabular halign="center">
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<row bottom="minor">
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<cell><m>x</m></cell>
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<cell><m>\Pr(x)</m></cell>
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</row>
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<row>
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<cell>1</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell>2</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell>3</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell>4</cell>
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<cell>0.2</cell>
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</row>
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<row>
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<cell>5</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell>6</cell>
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<cell>0.4</cell>
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</row>
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</tabular>
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</table>
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<p>
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With this distribution, if <m>A = \{2, 4, 6\}</m>, then:
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<md>
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<mrow> \Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}. </mrow>
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</md>
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</p>
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</statement>
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</example>
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<example>
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<statement>
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<p>
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Pick a number from 1 to 10.
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What is the probability of picking 3?
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</p>
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<p>
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Some considerations:
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<ol>
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<li>
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<p>
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Are we using the uniform distribution?
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</p>
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</li>
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<li>
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<p>
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What even is the sample space here? Do we need to pick only integers, or is <m>\pi</m> an outcome here?
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</p>
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</li>
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<li>
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<p>
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What would "uniform" mean if the sample space was infinite?
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</p>
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</li>
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</ol>
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</p>
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</statement>
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</example>
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<definition xml:id="def-probability-distribution">
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<statement>
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<p>
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A <term>probability distribution</term> on a sample space <m>\Omega</m> assigns probabilities to every event, satisfying the following conditions:
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<ol>
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<li>
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<p>
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<m>\Pr(\Omega) = 1</m>.
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</p>
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</li>
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<li>
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<p>
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<m>0 \leq \Pr(A) \leq 1</m> for any event <m>A</m>.
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</p>
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</li>
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<li>
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<p>
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If <m>A \cap B = \emptyset</m>, then <m>\Pr(A\cup B) = \Pr(A) + \Pr(B)</m>.
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</p>
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</li>
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</ol>
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</p>
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</statement>
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</definition>
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<example>
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<statement>
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<p>
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Suppose we flip a coin until we see heads.
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The sample space is <m>\Omega = \{H, TH, TTH, TTTH, \dotsc\}</m>.
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What would "uniform" mean here? Is there any way we could assign the same probability to each individual outcome here?
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</p>
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<p>
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Instead of "uniform", what if we want to treat the coin as a fair coin? Then what would the distribution be?
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</p>
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<table>
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<title>Distribution assuming a fair coin</title>
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<tabular halign="center">
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<row bottom="minor">
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<cell><m>x</m></cell>
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<cell># flips</cell>
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<cell><m>\Pr(x)</m></cell>
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</row>
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<row>
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<cell><m>H</m></cell>
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<cell>1</cell>
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<cell><m>1/2</m></cell>
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</row>
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<row>
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<cell><m>TH</m></cell>
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<cell>2</cell>
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<cell><m>1/4</m></cell>
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</row>
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<row>
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<cell><m>TTH</m></cell>
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<cell>3</cell>
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<cell><m>1/8</m></cell>
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</row>
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<row>
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<cell><m>TTTH</m></cell>
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<cell>4</cell>
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<cell><m>1/16</m></cell>
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</row>
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<row>
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<cell><m>\vdots</m></cell>
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<cell></cell>
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<cell><m>\vdots</m></cell>
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</row>
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<row>
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<cell><m>T\dotsm TH</m></cell>
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<cell><m>n</m></cell>
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<cell><m>1/2^n</m></cell>
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</row>
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<row>
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<cell><m>\vdots</m></cell>
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<cell></cell>
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<cell><m>\vdots</m></cell>
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</row>
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</tabular>
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</table>
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</statement>
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</example>
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<example>
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<statement>
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<p>
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One last (very useful!) observation.
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Recall: If <m>A\cap B = \emptyset</m>, then <m>\Pr(A\cup B) = \Pr(A) + \Pr(B)</m>.
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</p>
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<p>
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For any event <m>A</m>, <m>A\cap A^c = \emptyset</m> and <m>A \cup A^c = \Omega</m>.
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So:
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<md>
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<mrow> 1 = \Pr(\underbrace{A \cup A^c}_{\Omega}) = \Pr(A) + \Pr(A^c). </mrow>
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</md>
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We can rewrite this in two useful ways:
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<md>
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<mrow> \Pr(A) \amp = 1 - \Pr(A^c) </mrow>
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<mrow> \Pr(A^c) \amp = 1 - \Pr(A) </mrow>
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</md>
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</p>
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</statement>
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</example>
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</subsubsection>
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</subsection>
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<!--
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<subsection>
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<title>Thursday 1/15</title>
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<p>
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</p>
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</subsection>
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-->
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</section> |