More text and answers in Sec 1.1 and 1.2.
This commit is contained in:
@@ -146,8 +146,13 @@
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<css theme="boulder"/>
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<!-- <css theme="salem"/> -->
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<css
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theme="boulder"
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primary-color="#386F94"
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primary-color-dark="#5192BD"
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provide-dark-mode="yes"
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/>
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<!-- <css theme="salem"/> -->
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<!-- Search can be default or none, or you can use a Google-cx number to use google's search
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feature (but then variant should be set to none to avoid conflict) -->
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@@ -6,7 +6,6 @@
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<introduction>
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<p>
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Testing locally
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</p>
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</introduction>
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+331
-9
@@ -5,13 +5,20 @@
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<introduction>
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<p>
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Text before the first section.
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This chapter establishes the basic terminology of sample spaces, events, probability, conditi8onal probability, and independence.
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This language lets us model and talk about situations that involve randomness (or lack of knowledge which can sometimes be mathematically modeled in the same way).
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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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When we perform an experiment, there are many results that we might see.
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We want to be able to quantify the likelihood of seeing certain results.
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For this, we need to develop some mathematical terminology.
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</p>
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<definition xml:id="def-sample-space">
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<statement>
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<p>
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@@ -21,7 +28,7 @@
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</statement>
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</definition>
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<example>
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<example xml:id="example-sample-space">
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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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@@ -44,6 +51,7 @@
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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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Note that, for example, <m>(1, 2)</m> is a different outcome from <m>(2, 1)</m>.
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</p>
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</answer>
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</example>
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@@ -58,6 +66,10 @@
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</statement>
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</definition>
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<p>
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The subset symbol includes the possibility that <m>A</m> and <m>B</m> are equal sets, i.e., that they contain precisely the same elements.
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</p>
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<definition xml:id="def-set-operations">
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<statement>
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<p>
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@@ -108,8 +120,38 @@
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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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It's useful sometimes to draw pictures called <term>Venn diagrams</term> representing sets:
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</p>
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<figure xml:id="fig-Venn-diagram">
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<caption>Example Venn Diagram</caption>
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<image width="50%">
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<description>
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<p>
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Venn diagram showing sets <m>A, B, C</m> with the region representing <m>(A\cup B\cup C) - (A \cap C)</m> shaded.
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(90:1.75cm) circle (2.5cm)}
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\def\secondcircle{(210:1.75cm) circle (2.5cm)}
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\def\thirdcircle{(330:1.75cm) circle (2.5cm)}
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\fill[gray!30] \firstcircle;
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\fill[gray!30] \secondcircle;
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\fill[gray!30] \thirdcircle;
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\begin{scope}
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\clip \firstcircle;
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\clip \thirdcircle;
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\fill[white] \firstcircle;
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\end{scope}
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\draw \firstcircle node[text=black,above] {$A$};
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\draw \secondcircle node [text=black,below left] {$B$};
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\draw \thirdcircle node [text=black,below right] {$C$};
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\node at (0, -4.5) {$(A\cup B\cup C) - (A \cap C)$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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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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@@ -185,11 +227,27 @@
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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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Suppose we have a 6-sided die that's weighted to roll a 6 half of the time.
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We roll the 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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<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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Note that <m>\Omega</m> simply lists outcomes with no reference to the probabilities.
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So the answer here is the same as in <xref ref="example-sample-space"/>.
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</p>
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</answer>
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</exercise>
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<exercise>
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@@ -200,6 +258,31 @@
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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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<answer>
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<p>
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For two flips: <m>\Omega = \{ HH, HT, TH, TT \}</m>.
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</p>
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<p>
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For three flips: <m>\Omega = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}</m>.
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</p>
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<p>
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For four flips:
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<md>
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<mrow> \Omega = \{ \amp HHHH, HHHT, HHTH, HTHH, </mrow>
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<mrow> \amp THHH, HHTT, HTHT, HTTH, </mrow>
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<mrow> \amp THHT, THTH, TTHH, HTTT, </mrow>
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<mrow> \amp THTT, TTHT, TTTH, TTTT \}. </mrow>
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</md>
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</p>
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<p>
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Each additional flip doubles the number of outcomes.
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So, with ten flips, we'll have <m>|\Omega| = 2^{10} = 1024</m>.
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</p>
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</answer>
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</exercise>
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<exercise>
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@@ -208,6 +291,13 @@
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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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<answer>
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<p>
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Each additional roll will multiply the number of outcomes by 6.
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So, with 10 rolls, we'll have <m>|\Omega| = 6^{10}.</m>
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</p>
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</answer>
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</exercise>
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</exercises>
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</section>
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@@ -215,6 +305,10 @@
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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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Now that we have the language to refer to outcomes and events of an experiment, we want to start quantifying how likely those outcomes/events are to occur.
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</p>
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<definition xml:id="def-probability-distribution">
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<statement>
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<p>
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@@ -316,6 +410,11 @@
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</statement>
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</definition>
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<p>
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When we talk about a fair coin flip or a fair die roll, the word "fair" is indicating a uniform distribution.
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However, don't make the mistake of assuming that all distributions are uniform by default.
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</p>
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<example>
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<statement>
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<p>
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@@ -323,7 +422,17 @@
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What is the probability that they picked 3?
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</p>
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</statement>
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<!-- ehhhh? -->
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<answer>
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<p>
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Without assuming the distribution is fair (i.e., that each value <m>1, 2, \dotsc, 10</m>) has probability <m>1/10</m> of occurring), we don't have enough information to answer this question.
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</p>
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<p>
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In fact, the situation is even more vague than that: the sample space itself is unclear.
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Are we only allowed to pick integer values? What about fractions like <m>7/2</m>? What about irrational numbers like <m>\pi</m>?
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</p>
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</answer>
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</example>
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<exercises xml:id="exercises-Probability">
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@@ -389,8 +498,13 @@
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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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<answer>
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<p>
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<m>\Pr(A) = 0.45, \Pr(B) = 0.4, \Pr(A \cup B) = 0.6, \Pr(A \cap B) = 0.25.</m>
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</p>
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</answer>
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</exercise>
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<exercise>
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@@ -409,6 +523,12 @@
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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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<answer>
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<p>
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<m>\Omega = \{HH, HT, TH, TT\}.</m>
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</p>
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</answer>
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</task>
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@@ -418,6 +538,39 @@
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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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<answer>
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<table>
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<title>Probability Distribution for Two Fair Coin Flips</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><m>HH</m></cell>
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<cell>0.25</cell>
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</row>
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<row>
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<cell><m>HT</m></cell>
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<cell>0.25</cell>
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</row>
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<row>
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<cell><m>TH</m></cell>
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<cell>0.25</cell>
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</row>
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<row>
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<cell><m>TT</m></cell>
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<cell>0.25</cell>
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</row>
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</tabular>
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</table>
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</answer>
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</task>
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@@ -427,6 +580,39 @@
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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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<answer>
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<table>
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<title>Probability Distribution for Two Fair Coin Flips</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><m>HH</m></cell>
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<cell>0.09</cell>
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</row>
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<row>
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<cell><m>HT</m></cell>
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<cell>0.21</cell>
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</row>
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<row>
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<cell><m>TH</m></cell>
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<cell>0.21</cell>
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</row>
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<row>
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<cell><m>TT</m></cell>
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<cell>0.49</cell>
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</row>
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</tabular>
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</table>
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</answer>
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</task>
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</exercise>
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@@ -445,6 +631,19 @@
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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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<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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</task>
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@@ -454,6 +653,13 @@
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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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<answer>
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<p>
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We'll avoid an overly large table and note that, since the die is fair, every outcome is equally likely.
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Therefore, <m>\Pr(x) = \frac{1}{36}</m> for every <m>x\in \Omega</m>.
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</p>
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</answer>
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</task>
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@@ -461,9 +667,27 @@
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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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Find <m>\Pr(A), \Pr(B)</m>, and <m>\Pr(A \cap B)</m>.
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||||
</p>
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||||
</statement>
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||||
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||||
<answer>
|
||||
<p>
|
||||
<md>
|
||||
<mrow> A = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), </mrow>
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||||
<mrow> \amp (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
|
||||
<mrow> \amp (3, 4), (3, 5), (3, 6), </mrow>
|
||||
<mrow> \amp (4, 5), (4, 6), </mrow>
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<mrow> \amp (5, 6)\} </mrow>
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||||
<mrow> B = \{ \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
|
||||
<mrow> \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), </mrow>
|
||||
<mrow> \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
|
||||
<mrow> A \cap B = \{ \amp (2, 3), (2, 4), (2, 5), (2, 6), </mrow>
|
||||
<mrow> \amp (4, 5), (4, 6)\} </mrow>
|
||||
</md>
|
||||
Therefore <m>\Pr(A) = \frac{15}{36} = \frac{5}{12}, \Pr(B) = \frac{18}{36} = \frac{1}{2}, \Pr(A \cap B) = \frac{6}{36} = \frac{1}{6}.</m>
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||||
</p>
|
||||
</answer>
|
||||
</task>
|
||||
</exercise>
|
||||
|
||||
@@ -476,6 +700,50 @@
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||||
Write a probability distribution table for this die.
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<table>
|
||||
<title>Probability Distribution for a Linearly Scaled Die</title>
|
||||
|
||||
<tabular halign="center">
|
||||
<row bottom="minor">
|
||||
<cell><m>x</m></cell>
|
||||
<cell><m>\Pr(x)</m></cell>
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||||
</row>
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||||
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||||
<row>
|
||||
<cell>1</cell>
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||||
<cell><m>1/21</m></cell>
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||||
</row>
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||||
|
||||
<row>
|
||||
<cell>2</cell>
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||||
<cell><m>2/21</m></cell>
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||||
</row>
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||||
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||||
<row>
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||||
<cell>3</cell>
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||||
<cell><m>3/21</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>4/21</m></cell>
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||||
</row>
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||||
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||||
<row>
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||||
<cell>5</cell>
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||||
<cell><m>5/21</m></cell>
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||||
</row>
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||||
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||||
<row>
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||||
<cell>6</cell>
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||||
<cell><m>6/21</m></cell>
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||||
</row>
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||||
|
||||
</tabular>
|
||||
</table>
|
||||
</answer>
|
||||
</exercise>
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||||
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||||
<exercise>
|
||||
@@ -486,6 +754,49 @@
|
||||
Write a probability distribution table for this die.
|
||||
</p>
|
||||
</statement>
|
||||
<answer>
|
||||
<table>
|
||||
<title>Probability Distribution for an Even-biased Die</title>
|
||||
|
||||
<tabular halign="center">
|
||||
<row bottom="minor">
|
||||
<cell><m>x</m></cell>
|
||||
<cell><m>\Pr(x)</m></cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell>1</cell>
|
||||
<cell><m>1/9</m></cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell>2</cell>
|
||||
<cell><m>2/9</m></cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell>3</cell>
|
||||
<cell><m>1/9</m></cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell>4</cell>
|
||||
<cell><m>2/9</m></cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell>5</cell>
|
||||
<cell><m>1/9</m></cell>
|
||||
</row>
|
||||
|
||||
<row>
|
||||
<cell>6</cell>
|
||||
<cell><m>2/9</m></cell>
|
||||
</row>
|
||||
|
||||
</tabular>
|
||||
</table>
|
||||
</answer>
|
||||
</exercise>
|
||||
|
||||
<exercise>
|
||||
@@ -496,16 +807,27 @@
|
||||
What is the probability of the molecule leaving the cell during the first 3 minutes?
|
||||
</p>
|
||||
</statement>
|
||||
|
||||
<answer>
|
||||
<p>
|
||||
For short, write <m>\Pr(n)</m> to mean the probability of the toxin molecule leaving during the <m>n</m>th minute.
|
||||
Then <m>\Pr(n) = (0.7)^{n - 1} (0.3).</m>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The probability of leaving during the first 3 minutes is <m>\Pr(1) + \Pr(2) + \Pr(3) = 0.657.</m>
|
||||
</p>
|
||||
</answer>
|
||||
</exercise>
|
||||
|
||||
<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>
|
||||
</exercise> needs binomial coefficients -->
|
||||
</exercises>
|
||||
</section>
|
||||
|
||||
|
||||
+6
-1
@@ -40,7 +40,12 @@
|
||||
|
||||
<!-- If you put any latex-image elements you can include preambles -->
|
||||
<!-- for those in the next element. -->
|
||||
<latex-image-preamble> \usepackage{tikz, pgfplots} \usetikzlibrary{positioning,matrix,arrows} \usetikzlibrary{shapes,decorations,shadows,fadings,patterns} \usetikzlibrary{decorations.markings} </latex-image-preamble>
|
||||
<latex-image-preamble>
|
||||
\usepackage{tikz, pgfplots}
|
||||
\usetikzlibrary{positioning,matrix,arrows,calc}
|
||||
\usetikzlibrary{shapes,decorations,shadows,fadings,patterns}
|
||||
\usetikzlibrary{decorations.markings,decorations.pathreplacing}
|
||||
</latex-image-preamble>
|
||||
|
||||
<!-- It is possible to rename elements: -->
|
||||
<!-- <rename element="assemblage" xml:lang="en-US">Summary</rename> -->
|
||||
|
||||
Reference in New Issue
Block a user