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This commit is contained in:
andyeisenberg
@@ -121,11 +121,11 @@
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<!-- Whether to knowl a particular elements is set here -->
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<!-- Lots of elements have this possibility; see the guide -->
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<!--<knowl
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<knowl
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theorem="no"
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proof="yes"
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definition="no"
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example="yes"
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example="no"
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example-solution="yes"
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project="no"
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task="no"
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@@ -140,7 +140,7 @@
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exercise-divisional="no"
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exercise-worksheet="no"
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exercise-readingquestion="no"
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/>-->
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/>
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<!-- Specify the theme for the html by giving names to -->
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<!-- override defaults. Ex:<css theme="denver"/> -->
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+533
-397
@@ -1,411 +1,547 @@
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<?xml version="1.0" encoding="UTF-8"?>
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<chapter xml:id="ch-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Probability Theory</title>
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<title>Probability Theory</title>
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<introduction>
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<introduction>
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<p>
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Text before the first section.
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</p>
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</introduction>
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|
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<section xml:id="sec-Set-Theory" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Set Theory</title>
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<definition xml:id="def-sample-space">
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<statement>
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<p>
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Text before the first section.
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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>
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<statement>
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<p>
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An experiment consists of rolling a standard 6-sided die.
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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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<p>
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What would the sample space look like if we roll the die two times and recorded the results?
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</p>
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</statement>
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<answer>
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<p>
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<md>
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<mrow> \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </mrow>
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<mrow> \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
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<mrow> \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), </mrow>
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<mrow> \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </mrow>
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<mrow> \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), </mrow>
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<mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
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</md>
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</p>
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</answer>
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</example>
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<definition xml:id="def-subset">
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<statement>
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<p>
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Let <m>A</m> be a set.
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The symbol <m>\in</m> means "is an element of", as in <m>a \in A</m>.
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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>.
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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:
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</p>
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<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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|
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<li>
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<title>Intersection</title>
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|
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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>
|
||||
|
||||
<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>.
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</p>
|
||||
</li>
|
||||
|
||||
<li>
|
||||
<title>Complement</title>
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||||
|
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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>
|
||||
</li>
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||||
|
||||
<li>
|
||||
<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>
|
||||
</dl>
|
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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 representing sets...
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</p>
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<!-- TODO learn figures -->
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<definition xml:id="def-disjoint">
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<statement>
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<p>
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Two sets <m>A</m> and <m>B</m> are <term>disjoint</term> if <m>A \cap B = \emptyset</m>.
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</p>
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</statement>
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</definition>
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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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|
||||
<exercises xml:id="exercises-Set-Theory">
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<exercise>
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<introduction>
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<p>
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Consider the sets <m>A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</m>, <m>B = \{2, 4, 9, 10, 12, 14, 19\}</m>, and <m>C = \{9, 10, 11, 14, 16, 17, 20\}</m>, which are all subsets of <m>\Omega = \{1, 2, 3, \dotsc, 20\}</m>.
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</p>
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</introduction>
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||||
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<task>
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||||
<statement>
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||||
<p>
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||||
Find <m>A - (B \cap C)</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>\{1, 2, 3, 4, 5, 6, 7, 8\}</m>
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</p>
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||||
</answer>
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||||
</task>
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||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
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||||
Find <m>|A|</m>, <m>|B|</m>, <m>|C|</m>, <m>|A\cup B|</m>, <m>|A \cap B|</m>, <m>|B\cap C|</m>, <m>|A\cap C|</m>, and <m>|A\cup B\cup C|</m>.
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Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets?
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</p>
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</statement>
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<answer>
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<p>
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<m>|A| = 10</m>, <m>|B| = 7</m>, <m>|C| = 7</m>, <m>|A \cup B| = 13</m>, <m>|A \cap B| = 4</m>, <m>|B\cap C| = 3</m>, <m>|A\cap C| = 2</m>, <m>|A\cup B\cup C| = 17</m>. In particular, note that <m>|A\cup B| = 13 \neq 10 + 7 = |A| + |B|</m>, so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets.
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</p>
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||||
</answer>
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||||
</task>
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||||
|
||||
|
||||
<task>
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||||
<statement>
|
||||
<p>
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Find <m>A^c</m> and <m>(A\cup B)^c</m>.
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</p>
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||||
</statement>
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||||
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<answer>
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<p>
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<m>A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}</m>, <m>(A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}</m>.
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</p>
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||||
</answer>
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||||
</task>
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||||
</exercise>
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||||
<exercise>
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||||
<statement>
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||||
<p>
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||||
Suppose we roll a 6-sided die two times.
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List the set of all possible results.
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[Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result <m>(4, 2)</m>---rolling a 4 and then a 2.]
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||||
</p>
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||||
</statement>
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||||
</exercise>
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||||
|
||||
<exercise>
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||||
<statement>
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||||
<p>
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||||
Suppose we flip a coin two times.
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List the set of all possible results.
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What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be?
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||||
</p>
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||||
</statement>
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||||
</exercise>
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||||
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||||
<exercise>
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||||
<statement>
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||||
<p>
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||||
If we roll a 6-sided die ten times, how many possible results will there be?
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</p>
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||||
</statement>
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||||
</exercise>
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||||
</exercises>
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||||
</section>
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||||
|
||||
<section xml:id="sec-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
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||||
<title>Definition of Probability</title>
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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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</p>
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||||
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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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||||
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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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||||
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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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||||
</statement>
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||||
</definition>
|
||||
|
||||
<exercises xml:id="exercises-Probability">
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||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Consider the sample space <m>\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}</m> with probability distribution below.
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Calculate the probabilities of <m>A = \{1, 3, 7, 8\}</m>, <m>B = \{2, 3, 6, 7\}</m>, <m>A\cup B</m>, and <m>A \cap B</m>.
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</p>
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|
||||
<table>
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<title></title>
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||||
<tabular>
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||||
<row>
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||||
<cell halign="center"><m>x</m></cell>
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<cell halign="center">1</cell>
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<cell halign="center">2</cell>
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<cell halign="center">3</cell>
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<cell halign="center">4</cell>
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<cell halign="center">5</cell>
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<cell halign="center">6</cell>
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<cell halign="center">7</cell>
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<cell halign="center">8</cell>
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<cell halign="center">9</cell>
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</row>
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||||
|
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<row>
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<cell halign="center"><m>\Pr(x)</m></cell>
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<cell halign="center">0.1</cell>
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<cell halign="center">0.05</cell>
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<cell halign="center">0.2</cell>
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<cell halign="center">0.15</cell>
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||||
<cell halign="center">0.15</cell>
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||||
<cell halign="center">0.1</cell>
|
||||
<cell halign="center">0.05</cell>
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||||
<cell halign="center">0.1</cell>
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||||
<cell halign="center">0.1</cell>
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||||
</row>
|
||||
</tabular>
|
||||
</table>
|
||||
<!--</div attr= class="center">-->
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
Suppose we flip a coin two times.
|
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Answer the questions below.
|
||||
What about three flips? What about four flips?
|
||||
</p>
|
||||
</introduction>
|
||||
|
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|
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<task>
|
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<statement>
|
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<p>
|
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Write all outcomes in the sample space <m>\Omega</m>.
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</p>
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||||
</statement>
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||||
</task>
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|
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<task>
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<statement>
|
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<p>
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Make a probability distribution table for <m>\Omega</m> assuming the coin is fair.
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||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Make a probability distribution table assuming the coin comes up heads with probability 0.3.
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||||
</p>
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||||
</statement>
|
||||
</task>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
Suppose we roll a die two times.
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Answer the questions below.
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||||
</p>
|
||||
</introduction>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Write all outcomes in the sample space <m>\Omega</m>.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Make a probability distribution table for <m>\Omega</m> assuming the die is fair.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Let <m>A</m> be the event that the second roll is higher than the first, and let <m>B</m> be the event that the first roll is even.
|
||||
Find <m>\Pr(A), \Pr(B)</m>, and <m>\Pr(A \mid B)</m>.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Suppose a die has the values <m>1, 2, 3, 4, 5, 6</m> on the faces, but the die is not fair.
|
||||
Instead, the probabilities scale by the same amount as the face values.
|
||||
For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on.
|
||||
Write a probability distribution table for this die.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Suppose a die has the values <m>1, 2, 3, 4, 5, 6</m> on the faces, but the die is not fair.
|
||||
Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll.
|
||||
Write a probability distribution table for this die.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
|
||||
For each value of <m>n = 1, 2, 3, \dotsc</m>, find the probability of the toxin molecule leaving the cell during the <m>n</m>th minute.
|
||||
What is the probability of the molecule leaving the cell during the first 3 minutes?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
|
||||
For each value of <m>n = 1, 2, 3, \dotsc</m>, and for each value of <m>0\leq k \leq n</m>, find the probability that exactly <m>k</m> toxin molecules remain in the cell after the <m>n</m>th minute.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
|
||||
<section xml:id="sec-Conditional-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
|
||||
<title>Conditional Probability</title>
|
||||
|
||||
<p>
|
||||
Text of section.
|
||||
</p>
|
||||
|
||||
<exercises xml:id="exercises-Conditional-Probability"> <exercisegroup>
|
||||
<introduction>
|
||||
<p>
|
||||
In each of the following scenarios with given events <m>A</m> and <m>B</m>, alculate <m>\Pr(A), \Pr(B)</m>, <m>\Pr(A\cap B)</m>, <m>\Pr(A \mid B)</m>, and <m>\Pr(B \mid A)</m>.
|
||||
</p>
|
||||
</introduction>
|
||||
|
||||
<section xml:id="sec-Set-Theory" xmlns:xi="http://www.w3.org/2001/XInclude">
|
||||
<title>Set Theory</title>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Text of section.
|
||||
An experiment consists of rolling a fair die two times.
|
||||
Let <m>A</m> be the event that the sum is even, and let <m>B</m> be the event that the second roll is higher than the first.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of flipping a fair coin three times.
|
||||
Let <m>A</m> be the event that the first and second flips match.
|
||||
Let <m>B</m> be the event that there are at least two heads.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercisegroup>
|
||||
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
A diagnostic test is developed to detect a disease present in 3.2% of the population.
|
||||
For a patient who has the disease, the test will accurately give a positive result 65% of the time.
|
||||
When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.
|
||||
</p>
|
||||
</introduction>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
For a patient who receives a positive test, what is the probability they have the disease?
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
TODO
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
For a patient who receives a negative test, what is the probability they do not have the disease?
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
TODO
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
|
||||
<section xml:id="sec-Independent-Events" xmlns:xi="http://www.w3.org/2001/XInclude">
|
||||
<title>Independent Events</title>
|
||||
|
||||
<p>
|
||||
Text of section.
|
||||
</p>
|
||||
|
||||
<exercises xml:id="exercises-Independent-Events">
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of rolling a fair die two times.
|
||||
Let <m>A</m> be the event that the sum is even, and let <m>B</m> be the event that the second roll is higher than the first.
|
||||
Are <m>A</m> and <m>B</m> independent?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of flipping a fair coin three times.
|
||||
Let <m>A</m> be the event that the first and second flips match.
|
||||
Let <m>B</m> be the event that there are at least two heads.
|
||||
Are <m>A</m> and <m>B</m> independent?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Let <m>A = \{1, 2, 3\}</m> and <m>B = \{3, 4, 5\}</m>.
|
||||
Fill in the following probability distribution table so that <m>A, B</m> are independent.
|
||||
</p>
|
||||
|
||||
<exercises xml:id="exercises-Set-Theory">
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
Consider the sets <m>A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</m>, <m>B = \{2, 4, 9, 10, 12, 14, 19\}</m>, and <m>C = \{9, 10, 11, 14, 16, 17, 20\}</m>, which are all subsets of <m>\Omega = \{1, 2, 3, \dotsc, 20\}</m>.
|
||||
</p>
|
||||
</introduction>
|
||||
<table>
|
||||
<title></title>
|
||||
|
||||
<tabular>
|
||||
<row>
|
||||
<cell halign="center"><m>x</m></cell>
|
||||
<cell halign="center">1</cell>
|
||||
<cell halign="center">2</cell>
|
||||
<cell halign="center">3</cell>
|
||||
<cell halign="center">4</cell>
|
||||
<cell halign="center">5</cell>
|
||||
<cell halign="center">6</cell>
|
||||
</row>
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Find <m>A - (B \cap C)</m>.
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
<m>\{1, 2, 3, 4, 5, 6, 7, 8\}</m>
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Find <m>|A|</m>, <m>|B|</m>, <m>|C|</m>, <m>|A\cup B|</m>, <m>|A \cap B|</m>, <m>|B\cap C|</m>, <m>|A\cap C|</m>, and <m>|A\cup B\cup C|</m>.
|
||||
Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets?
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
<m>|A| = 10</m>, <m>|B| = 7</m>, <m>|C| = 7</m>, <m>|A \cup B| = 13</m>, <m>|A \cap B| = 4</m>, <m>|B\cap C| = 3</m>, <m>|A\cap C| = 2</m>, <m>|A\cup B\cup C| = 17</m>. In particular, note that <m>|A\cup B| = 13 \neq 10 + 7 = |A| + |B|</m>, so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets.
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Find <m>A^c</m> and <m>(A\cup B)^c</m>.
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
<m>A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}</m>, <m>(A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}</m>.
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Suppose we roll a 6-sided die two times.
|
||||
List the set of all possible results.
|
||||
[Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result <m>(4, 2)</m>---rolling a 4 and then a 2.]
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Suppose we flip a coin two times.
|
||||
List the set of all possible results.
|
||||
What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
If we roll a 6-sided die ten times, how many possible results will there be?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
|
||||
<section xml:id="sec-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
|
||||
<title>Definition of Probability</title>
|
||||
|
||||
<p>
|
||||
Text of section.
|
||||
</p>
|
||||
|
||||
<exercises xml:id="exercises-Probability">
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Consider the sample space <m>\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}</m> with probability distribution below.
|
||||
Calculate the probabilities of <m>A = \{1, 3, 7, 8\}</m>, <m>B = \{2, 3, 6, 7\}</m>, <m>A\cup B</m>, and <m>A \cap B</m>.
|
||||
</p>
|
||||
|
||||
<table>
|
||||
<title></title>
|
||||
|
||||
<tabular>
|
||||
<row>
|
||||
<cell halign="center"><m>x</m></cell>
|
||||
<cell halign="center">1</cell>
|
||||
<cell halign="center">2</cell>
|
||||
<cell halign="center">3</cell>
|
||||
<cell halign="center">4</cell>
|
||||
<cell halign="center">5</cell>
|
||||
<cell halign="center">6</cell>
|
||||
<cell halign="center">7</cell>
|
||||
<cell halign="center">8</cell>
|
||||
<cell halign="center">9</cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell halign="center"><m>\Pr(x)</m></cell>
|
||||
<cell halign="center">0.1</cell>
|
||||
<cell halign="center">0.05</cell>
|
||||
<cell halign="center">0.2</cell>
|
||||
<cell halign="center">0.15</cell>
|
||||
<cell halign="center">0.15</cell>
|
||||
<cell halign="center">0.1</cell>
|
||||
<cell halign="center">0.05</cell>
|
||||
<cell halign="center">0.1</cell>
|
||||
<cell halign="center">0.1</cell>
|
||||
</row>
|
||||
</tabular>
|
||||
</table>
|
||||
<!--</div attr= class="center">-->
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
Suppose we flip a coin two times.
|
||||
Answer the questions below.
|
||||
What about three flips? What about four flips?
|
||||
</p>
|
||||
</introduction>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Write all outcomes in the sample space <m>\Omega</m>.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Make a probability distribution table for <m>\Omega</m> assuming the coin is fair.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Make a probability distribution table assuming the coin comes up heads with probability 0.3.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
Suppose we roll a die two times.
|
||||
Answer the questions below.
|
||||
</p>
|
||||
</introduction>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Write all outcomes in the sample space <m>\Omega</m>.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Make a probability distribution table for <m>\Omega</m> assuming the die is fair.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
Let <m>A</m> be the event that the second roll is higher than the first, and let <m>B</m> be the event that the first roll is even.
|
||||
Find <m>\Pr(A), \Pr(B)</m>, and <m>\Pr(A \mid B)</m>.
|
||||
</p>
|
||||
</statement>
|
||||
</task>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Suppose a die has the values <m>1, 2, 3, 4, 5, 6</m> on the faces, but the die is not fair.
|
||||
Instead, the probabilities scale by the same amount as the face values.
|
||||
For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on.
|
||||
Write a probability distribution table for this die.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Suppose a die has the values <m>1, 2, 3, 4, 5, 6</m> on the faces, but the die is not fair.
|
||||
Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll.
|
||||
Write a probability distribution table for this die.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
|
||||
For each value of <m>n = 1, 2, 3, \dotsc</m>, find the probability of the toxin molecule leaving the cell during the <m>n</m>th minute.
|
||||
What is the probability of the molecule leaving the cell during the first 3 minutes?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
|
||||
For each value of <m>n = 1, 2, 3, \dotsc</m>, and for each value of <m>0\leq k \leq n</m>, find the probability that exactly <m>k</m> toxin molecules remain in the cell after the <m>n</m>th minute.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
|
||||
<section xml:id="sec-Conditional-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
|
||||
<title>Conditional Probability</title>
|
||||
|
||||
<p>
|
||||
Text of section.
|
||||
</p>
|
||||
|
||||
<exercises xml:id="exercises-Conditional-Probability"> <exercisegroup>
|
||||
<introduction>
|
||||
<p>
|
||||
In each of the following scenarios with given events <m>A</m> and <m>B</m>, alculate <m>\Pr(A), \Pr(B)</m>, <m>\Pr(A\cap B)</m>, <m>\Pr(A \mid B)</m>, and <m>\Pr(B \mid A)</m>.
|
||||
</p>
|
||||
</introduction>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of rolling a fair die two times.
|
||||
Let <m>A</m> be the event that the sum is even, and let <m>B</m> be the event that the second roll is higher than the first.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of flipping a fair coin three times.
|
||||
Let <m>A</m> be the event that the first and second flips match.
|
||||
Let <m>B</m> be the event that there are at least two heads.
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercisegroup>
|
||||
|
||||
<exercise>
|
||||
<introduction>
|
||||
<p>
|
||||
A diagnostic test is developed to detect a disease present in 3.2% of the population.
|
||||
For a patient who has the disease, the test will accurately give a positive result 65% of the time.
|
||||
When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.
|
||||
</p>
|
||||
</introduction>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
For a patient who receives a positive test, what is the probability they have the disease?
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
TODO
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
|
||||
|
||||
<task>
|
||||
<statement>
|
||||
<p>
|
||||
For a patient who receives a negative test, what is the probability they do not have the disease?
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
TODO
|
||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
|
||||
<section xml:id="sec-Independent-Events" xmlns:xi="http://www.w3.org/2001/XInclude">
|
||||
<title>Independent Events</title>
|
||||
|
||||
<p>
|
||||
Text of section.
|
||||
</p>
|
||||
|
||||
<exercises xml:id="exercises-Independent-Events">
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of rolling a fair die two times.
|
||||
Let <m>A</m> be the event that the sum is even, and let <m>B</m> be the event that the second roll is higher than the first.
|
||||
Are <m>A</m> and <m>B</m> independent?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
An experiment consists of flipping a fair coin three times.
|
||||
Let <m>A</m> be the event that the first and second flips match.
|
||||
Let <m>B</m> be the event that there are at least two heads.
|
||||
Are <m>A</m> and <m>B</m> independent?
|
||||
</p>
|
||||
</statement>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
<statement>
|
||||
<p>
|
||||
Let <m>A = \{1, 2, 3\}</m> and <m>B = \{3, 4, 5\}</m>.
|
||||
Fill in the following probability distribution table so that <m>A, B</m> are independent.
|
||||
</p>
|
||||
|
||||
<table>
|
||||
<title></title>
|
||||
|
||||
<tabular>
|
||||
<row>
|
||||
<cell halign="center"><m>x</m></cell>
|
||||
<cell halign="center">1</cell>
|
||||
<cell halign="center">2</cell>
|
||||
<cell halign="center">3</cell>
|
||||
<cell halign="center">4</cell>
|
||||
<cell halign="center">5</cell>
|
||||
<cell halign="center">6</cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell halign="center"><m>\Pr(x)</m></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
</row>
|
||||
</tabular>
|
||||
</table>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
<row>
|
||||
<cell halign="center"><m>\Pr(x)</m></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
<cell halign="center"></cell>
|
||||
</row>
|
||||
</tabular>
|
||||
</table>
|
||||
</statement>
|
||||
</exercise>
|
||||
</exercises>
|
||||
</section>
|
||||
</chapter>
|
||||
@@ -7,21 +7,21 @@
|
||||
<bibinfo>
|
||||
|
||||
<author>
|
||||
<personname>You</personname>
|
||||
<department>Your department</department>
|
||||
<institution>Your institution</institution>
|
||||
<personname>Andy Eisenberg</personname>
|
||||
<department>Mathematics Department</department>
|
||||
<institution>Temple University</institution>
|
||||
</author>
|
||||
<date>
|
||||
<today />
|
||||
</date>
|
||||
|
||||
<website>
|
||||
<!-- <website>
|
||||
<url href="https://pretextbook.org">My Website</url>
|
||||
</website>
|
||||
</website> -->
|
||||
|
||||
<copyright>
|
||||
<year>2020<ndash />2024</year>
|
||||
<holder>You</holder>
|
||||
<year>2025</year>
|
||||
<holder>Andy Eisenberg</holder>
|
||||
<shortlicense> This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit <url href="http://creativecommons.org/licenses/by-sa/4.0/" visual="creativecommons.org/licenses/by-sa/4.0"> CreativeCommons.org</url>
|
||||
</shortlicense>
|
||||
</copyright>
|
||||
|
||||
Reference in New Issue
Block a user