411 lines
15 KiB
XML
411 lines
15 KiB
XML
<?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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<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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<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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<p>
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Text of section.
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</p>
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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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<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>
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<statement>
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<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>
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<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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<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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<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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<p>
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Text of section.
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</p>
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<exercises xml:id="exercises-Probability">
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<exercise>
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<statement>
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<p>
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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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<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>
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<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>
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</tabular>
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</table>
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<!--</div attr= class="center">-->
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</statement>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Suppose we flip a coin two times.
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Answer the questions below.
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What about three flips? What about four flips?
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</p>
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</introduction>
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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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<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>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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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>
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</task>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Suppose we roll a die two times.
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Answer the questions below.
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</p>
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</introduction>
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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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<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 die is fair.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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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.
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Find <m>\Pr(A), \Pr(B)</m>, and <m>\Pr(A \mid B)</m>.
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</p>
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</statement>
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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 a die has the values <m>1, 2, 3, 4, 5, 6</m> on the faces, but the die is not fair.
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Instead, the probabilities scale by the same amount as the face values.
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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.
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Write a probability distribution table for this die.
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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 a die has the values <m>1, 2, 3, 4, 5, 6</m> on the faces, but the die is not fair.
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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.
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Write a probability distribution table for this die.
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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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A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
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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.
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What is the probability of the molecule leaving the cell during the first 3 minutes?
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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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Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
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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.
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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-Conditional-Probability" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Conditional Probability</title>
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<p>
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Text of section.
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</p>
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<exercises xml:id="exercises-Conditional-Probability"> <exercisegroup>
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<introduction>
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<p>
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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>.
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</p>
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</introduction>
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<exercise>
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<statement>
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<p>
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An experiment consists of rolling a fair die two times.
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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.
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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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An experiment consists of flipping a fair coin three times.
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Let <m>A</m> be the event that the first and second flips match.
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Let <m>B</m> be the event that there are at least two heads.
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</p>
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</statement>
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</exercise>
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</exercisegroup>
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<exercise>
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<introduction>
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<p>
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A diagnostic test is developed to detect a disease present in 3.2% of the population.
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For a patient who has the disease, the test will accurately give a positive result 65% of the time.
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When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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For a patient who receives a positive test, what is the probability they have the disease?
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</p>
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</statement>
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<answer>
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<p>
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TODO
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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>
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<p>
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For a patient who receives a negative test, what is the probability they do not have the disease?
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</p>
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</statement>
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<answer>
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<p>
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TODO
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</p>
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</answer>
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</task>
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</exercise>
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</exercises>
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</section>
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<section xml:id="sec-Independent-Events" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Independent Events</title>
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<p>
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Text of section.
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</p>
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<exercises xml:id="exercises-Independent-Events">
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<exercise>
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<statement>
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<p>
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An experiment consists of rolling a fair die two times.
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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.
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Are <m>A</m> and <m>B</m> independent?
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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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An experiment consists of flipping a fair coin three times.
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Let <m>A</m> be the event that the first and second flips match.
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Let <m>B</m> be the event that there are at least two heads.
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Are <m>A</m> and <m>B</m> independent?
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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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Let <m>A = \{1, 2, 3\}</m> and <m>B = \{3, 4, 5\}</m>.
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Fill in the following probability distribution table so that <m>A, B</m> are independent.
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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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</row>
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<row>
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<cell halign="center"><m>\Pr(x)</m></cell>
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<cell halign="center"></cell>
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<cell halign="center"></cell>
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<cell halign="center"></cell>
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<cell halign="center"></cell>
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<cell halign="center"></cell>
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<cell halign="center"></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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</exercise>
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</exercises>
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</section>
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</chapter> |