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<introduction>
<p>
This is an outline of the topics we covered in the first week of class.
These notes are <em>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.
</p>
</introduction>
<!-- Can also be <handout> instead of <subsection> to get a printable version -->
<subsection>
<title>Monday 8/22</title>
<title>Tuesday 1/13</title>
<p>
In calculus, you mostly asked <term>deterministic</term> questions about functions.
In science experiments, you need to take <term>randomness</term> into account.
</p>
<example>
<statement>
<p>
A toxin molecule in a cell has a certain chance each minute to leave the cell.
</p>
</statement>
</example>
<example>
<statement>
<p>
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.
</p>
</statement>
</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">
<statement>
<p>
The <term>sample space</term>, often denoted <m>\Omega</m>, is the set of all possible results of an experiment.
A single result is called an <term>outcome</term>, while a collection of results is called an <term>event</term>.
</p>
</statement>
</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">
<statement>
<p>
An experiment consists of rolling a standard 6-sided die.
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.
</p>
</statement>
</example>
<definition xml:id="def-set-operations">
<statement>
<p>
Consider sets <m>A</m> and <m>B</m>, each contained inside <m>\Omega</m>.
We can combine sets in a variety of ways: <dl>
<li>
<title>Union</title>
<p>
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>.
</p>
</li>
<li>
<title>Intersection</title>
<p>
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>.
</p>
</li>
<li>
<title>Difference</title>
<p>
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>.
</p>
</li>
<li>
<title>Complement</title>
<p>
The <term>complement</term> of <m>A</m> is the set <m>A^c = \{x \in \Omega \mid x \notin A\}</m>.
</p>
</li>
<li>
<title>Empty Set</title>
<p>
The <term>empty set</term>, usually written <m>\emptyset</m> or <m>\{\}</m>, is the set which contains no elements.
</p>
</li>
</dl>
</p>
</statement>
</definition>
<p>
It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing sets:
</p>
<figure xml:id="fig-Venn-diagram">
<caption>Example Venn Diagram</caption>
<image width="50%">
<description>
<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.
</p>
</description>
<latex-image>
\begin{tikzpicture}
\def\firstcircle{(90:1.75cm) circle (2.5cm)}
\def\secondcircle{(210:1.75cm) circle (2.5cm)}
\def\thirdcircle{(330:1.75cm) circle (2.5cm)}
\fill[gray!30] \firstcircle;
\fill[gray!30] \secondcircle;
\fill[gray!30] \thirdcircle;
\begin{scope}
\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}
</latex-image>
</image>
</figure>
<p>
Next, we want to start assigning probabilities to each individual outcome so we can then find the probabilities of events.
</p>
<example>
<statement>
<p>
An experiment consists of rolling a standard 6-sided die.
The sample space is <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
We might assign probabilities as follows:
</p>
<table>
<title>Distribution for a fair die</title>
<tabular halign="center">
<row bottom="minor">
<cell><m>x</m></cell>
<cell><m>\Pr(x)</m></cell>
</row>
<row>
<cell>1</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>2</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>3</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>4</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>5</cell>
<cell><m>1/6</m></cell>
</row>
<row>
<cell>6</cell>
<cell><m>1/6</m></cell>
</row>
</tabular>
</table>
<p>
Note that we don't have to assign the same probability to each outcome.
If we do, we call this distribution <term>uniform</term>.
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:
<md>
<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>
</md>
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>.
</p>
</statement>
</example>
<definition xml:id="def-cardinality">
<statement>
<p>
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>.
</p>
</statement>
</definition>
<fact>
<statement>
<p>
If <m>\Omega</m> is a finite probability space with the uniform distribution and <m>A \subset \Omega</m> is an event, then:
<md>
<mrow> \Pr(A) = \frac{|A|}{|\Omega|} </mrow>
</md>
</p>
</statement>
</fact>
<example>
<statement>
<p>
Suppose we have a weighted die that's much more likely to come up 6 than any other outcome.
</p>
<table>
<title>Distribution for a fair die</title>
<tabular halign="center">
<row bottom="minor">
<cell><m>x</m></cell>
<cell><m>\Pr(x)</m></cell>
</row>
<row>
<cell>1</cell>
<cell>0.1</cell>
</row>
<row>
<cell>2</cell>
<cell>0.1</cell>
</row>
<row>
<cell>3</cell>
<cell>0.1</cell>
</row>
<row>
<cell>4</cell>
<cell>0.2</cell>
</row>
<row>
<cell>5</cell>
<cell>0.1</cell>
</row>
<row>
<cell>6</cell>
<cell>0.4</cell>
</row>
</tabular>
</table>
<p>
With this distribution, if <m>A = \{2, 4, 6\}</m>, then:
<md>
<mrow> \Pr(A) = 0.1 + 0.2 + 0.4 = 0.7 \neq \frac{3}{6}. </mrow>
</md>
</p>
</statement>
</example>
</subsection>
<!--
<subsection>
<title>Thursday 1/15</title>
<p>
</p>
</subsection>
<subsection>
<title>Wednesday 8/24</title>
<p>
</p>
</subsection>
<subsection>
<title>Friday 8/26</title>
<p>
</p>
</subsection>
</section>
-->
</section>