Figured out the list error
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andyeisenberg
+172
-112
@@ -62,49 +62,48 @@
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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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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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<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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</statement>
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</definition>
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@@ -220,30 +219,93 @@
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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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<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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</statement>
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</definition>
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<p>
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For small probability spaces (i.e., with finitely many outcomes in the sample space), we'll usually assign probabilities to each individual outcome, and perhaps list them in a table.
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Then, to find the probability of any event, simply add together the probabilities of each outcome in that event.
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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 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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The probability distribution (assuming a fair die) is shown below.
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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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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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The probability of <m>A</m> is:
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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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</p>
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</statement>
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</example>
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<exercises xml:id="exercises-Probability">
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<exercise>
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<statement>
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@@ -255,31 +317,55 @@
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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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<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 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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<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>0.05</cell>
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</row>
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<row>
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<cell>3</cell>
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<cell>0.2</cell>
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</row>
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<row>
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<cell>4</cell>
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<cell>0.15</cell>
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</row>
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<row>
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<cell>5</cell>
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<cell>0.15</cell>
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</row>
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<row>
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<cell>6</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell>7</cell>
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<cell>0.05</cell>
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</row>
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<row>
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<cell>8</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell>9</cell>
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<cell>0.1</cell>
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</row>
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</tabular>
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</table>
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@@ -511,35 +597,9 @@
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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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Let <m>A = \{1, 2, 3\}</m> and <m>B = \{3, 4, 5\}</m> be events in the sample space <m>\Omega = \{1, 2, 3, 4, 5, 6\}</m>.
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Create a probability distribution for <m>\Omega</m> 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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