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<subsection> <subsection>
<title>Tuesday 1/13</title> <title>Tuesday 1/13</title>
<subsubsection xml:id="subsubsec-Sets">
<title>Sec 1.1: Sets</title>
<p> <p>
In calculus, you mostly asked <term>deterministic</term> questions about functions. In prior math courses, you mostly asked <term>deterministic</term> questions.
In science experiments, you need to take <term>randomness</term> into account. Now, we need new tools to model <term>randomness</term>.
</p> </p>
<example> <example>
@@ -37,14 +40,6 @@
</statement> </statement>
</example> </example>
<example>
<statement>
<p>
Based on historical year-over-year population growth rate data for a particular species, what is a reasonable range for next year's population?
</p>
</statement>
</example>
<definition xml:id="def-sample-space"> <definition xml:id="def-sample-space">
<statement> <statement>
<p> <p>
@@ -54,26 +49,34 @@
</statement> </statement>
</definition> </definition>
<definition xml:id="def-subset">
<statement>
<p>
Let <m>A</m> be a set.
The symbol <m>\in</m> means "is an element of", as in <m>a \in A</m>.
Given another set <m>B</m>, we say <m>A</m> is a <term>subset</term> of <m>B</m>, written <m>A\subset B</m>, to mean that every element of the set <m>A</m> is also an element of the set <m>B</m>.
</p>
</statement>
</definition>
<example xml:id="example-sample-space"> <example xml:id="example-sample-space">
<statement> <statement>
<p> <p>
An experiment consists of rolling a standard 6-sided die. An experiment consists of rolling a standard 6-sided die (D6).
The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>. The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the result of the roll is even. One possible event is <m>A = \{2, 4, 6\}</m>, i.e., the event that the result of the roll is even.
</p> </p>
</statement> </statement>
</example> </example>
<p>
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.
</p>
<definition xml:id="def-subset">
<statement>
<p>
The symbol <m>\in</m> means "is an <term>element</term> of", as in <m>4 \in A</m>.
</p>
<p>
The symbol <m>\subset</m> means "is a <term>subset</term> of", as in <m>A \subset \Omega</m>.
This means that every element of the set <m>A</m> is also an element of the set <m>\Omega</m>.
</p>
</statement>
</definition>
<definition xml:id="def-set-operations"> <definition xml:id="def-set-operations">
<statement> <statement>
<p> <p>
@@ -124,39 +127,107 @@
</definition> </definition>
<p> <p>
It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing sets: It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing set interactions.
</p> </p>
<figure xml:id="fig-Venn-diagram"> <sbsgroup widths="40% 40%">
<caption>Example Venn Diagram</caption> <sidebyside>
<image width="50%"> <figure xml:id="fig-Venn-diagram-union">
<caption>Union</caption>
<image>
<description> <description>
<p> <p>
Venn diagram showing sets <m>A, B, C</m> with the region representing <m>(A\cup B\cup C) - (A \cap C)</m> shaded. Venn diagram showing sets <m>A, B</m> with the region representing <m>A \cup B</m> shaded.
</p> </p>
</description> </description>
<latex-image> <latex-image>
\begin{tikzpicture} \begin{tikzpicture}
\def\firstcircle{(90:1.75cm) circle (2.5cm)} \def\firstcircle{(180:1.75cm) circle (2.5cm)}
\def\secondcircle{(210:1.75cm) circle (2.5cm)} \def\secondcircle{(0:1.75cm) circle (2.5cm)}
\def\thirdcircle{(330:1.75cm) circle (2.5cm)}
\fill[gray!30] \firstcircle; \fill[gray!30] \firstcircle;
\fill[gray!30] \secondcircle; \fill[gray!30] \secondcircle;
\fill[gray!30] \thirdcircle; \draw \firstcircle node[text=black,left] {$A$};
\begin{scope} \draw \secondcircle node [text=black,right] {$B$};
\clip \firstcircle;
\clip \thirdcircle;
\fill[white] \firstcircle;
\end{scope}
\draw \firstcircle node[text=black,above] {$A$};
\draw \secondcircle node [text=black,below left] {$B$};
\draw \thirdcircle node [text=black,below right] {$C$};
\node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$};
\end{tikzpicture} \end{tikzpicture}
</latex-image> </latex-image>
</image> </image>
</figure> </figure>
<figure xml:id="fig-Venn-diagram-intersection">
<caption>Intersection</caption>
<image>
<description>
<p>
Venn diagram showing sets <m>A, B</m> with the region representing <m>A \cap B</m> shaded.
</p>
</description>
<latex-image>
\begin{tikzpicture}
\def\firstcircle{(180:1.75cm) circle (2.5cm)}
\def\secondcircle{(0:1.75cm) circle (2.5cm)}
\fill[white] \firstcircle;
\fill[white] \secondcircle;
\begin{scope}
\clip \firstcircle;
\clip \secondcircle;
\fill[gray!30] \firstcircle;
\end{scope}
\draw \firstcircle node[text=black,left] {$A$};
\draw \secondcircle node [text=black,right] {$B$};
\end{tikzpicture}
</latex-image>
</image>
</figure>
</sidebyside>
<sidebyside>
<figure xml:id="fig-Venn-diagram-difference">
<caption>Difference</caption>
<image>
<description>
<p>
Venn diagram showing sets <m>A, B</m> with the region representing <m>A - B</m> shaded.
</p>
</description>
<latex-image>
\begin{tikzpicture}
\def\firstcircle{(180:1.75cm) circle (2.5cm)}
\def\secondcircle{(0:1.75cm) circle (2.5cm)}
\fill[gray!30] \firstcircle;
\fill[white] \secondcircle;
\draw \firstcircle node[text=black,left] {$A$};
\draw \secondcircle node [text=black,right] {$B$};
\end{tikzpicture}
</latex-image>
</image>
</figure>
<figure xml:id="fig-Venn-diagram-complement">
<caption>Complement</caption>
<image>
<description>
<p>
Venn diagram showing set <m>A \subset \Omega</m> with the region representing <m>A^c</m> shaded.
</p>
</description>
<latex-image>
\begin{tikzpicture}
\def\firstcircle{(0, 0) circle (2.5cm)}
\fill [gray!30] (-4, -3) rectangle (5, 3);
\fill[white] \firstcircle;
\draw \firstcircle node[text=black] {$A$};
\draw (-4, -3) rectangle (5, 3) node [text=black,right] {$\Omega$};
\end{tikzpicture}
</latex-image>
</image>
</figure>
</sidebyside>
</sbsgroup>
</subsubsection>
<subsubsection xml:id="subsubsec-Probability">
<title>Sec 1.2: Probability</title>
<p> <p>
Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events. Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.
</p> </p>
@@ -164,7 +235,7 @@
<example> <example>
<statement> <statement>
<p> <p>
An experiment consists of rolling a standard 6-sided die. An experiment consists of rolling a D6.
The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>. The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
We might assign probabilities as follows: We might assign probabilities as follows:
</p> </p>
@@ -296,6 +367,156 @@
</p> </p>
</statement> </statement>
</example> </example>
<example>
<statement>
<p>
Pick a number from 1 to 10.
What is the probability of picking 3?
</p>
<p>
Some considerations:
<ol>
<li>
<p>
Are we using the uniform distribution?
</p>
</li>
<li>
<p>
What even is the sample space here? Do we need to pick only integers, or is <m>\pi</m> an outcome here?
</p>
</li>
<li>
<p>
What would "uniform" mean if the sample space was infinite?
</p>
</li>
</ol>
</p>
</statement>
</example>
<definition xml:id="def-probability-distribution">
<statement>
<p>
A <term>probability distribution</term> on a sample space <m>\Omega</m> assigns probabilities to every event, satisfying the following conditions:
<ol>
<li>
<p>
<m>\Pr(\Omega) = 1</m>.
</p>
</li>
<li>
<p>
<m>0 \leq \Pr(A) \leq 1</m> for any event <m>A</m>.
</p>
</li>
<li>
<p>
If <m>A \cap B = \emptyset</m>, then <m>\Pr(A\cup B) = \Pr(A) + \Pr(B)</m>.
</p>
</li>
</ol>
</p>
</statement>
</definition>
<example>
<statement>
<p>
Suppose we flip a coin until we see heads.
The sample space is <m>\Omega = \{H, TH, TTH, TTTH, \dotsc\}</m>.
What would "uniform" mean here? Is there any way we could assign the same probability to each individual outcome here?
</p>
<p>
Instead of "uniform", what if we want to treat the coin as a fair coin? Then what would the distribution be?
</p>
<table>
<title>Distribution assuming a fair coin</title>
<tabular halign="center">
<row bottom="minor">
<cell><m>x</m></cell>
<cell># flips</cell>
<cell><m>\Pr(x)</m></cell>
</row>
<row>
<cell><m>H</m></cell>
<cell>1</cell>
<cell><m>1/2</m></cell>
</row>
<row>
<cell><m>TH</m></cell>
<cell>2</cell>
<cell><m>1/4</m></cell>
</row>
<row>
<cell><m>TTH</m></cell>
<cell>3</cell>
<cell><m>1/8</m></cell>
</row>
<row>
<cell><m>TTTH</m></cell>
<cell>4</cell>
<cell><m>1/16</m></cell>
</row>
<row>
<cell><m>\vdots</m></cell>
<cell></cell>
<cell><m>\vdots</m></cell>
</row>
<row>
<cell><m>T\dotsm TH</m></cell>
<cell><m>n</m></cell>
<cell><m>1/2^n</m></cell>
</row>
<row>
<cell><m>\vdots</m></cell>
<cell></cell>
<cell><m>\vdots</m></cell>
</row>
</tabular>
</table>
</statement>
</example>
<example>
<statement>
<p>
One last (very useful!) observation.
Recall: If <m>A\cap B = \emptyset</m>, then <m>\Pr(A\cup B) = \Pr(A) + \Pr(B)</m>.
</p>
<p>
For any event <m>A</m>, <m>A\cap A^c = \emptyset</m> and <m>A \cup A^c = \Omega</m>.
So:
<md>
<mrow> 1 = \Pr(\underbrace{A \cup A^c}_{\Omega}) = \Pr(A) + \Pr(A^c). </mrow>
</md>
We can rewrite this in two useful ways:
<md>
<mrow> \Pr(A) \amp = 1 - \Pr(A^c) </mrow>
<mrow> \Pr(A^c) \amp = 1 - \Pr(A) </mrow>
</md>
</p>
</statement>
</example>
</subsubsection>
</subsection> </subsection>
<!-- <!--
<subsection> <subsection>