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<quiz>
  <question type="stack">
    <name>
      <text>(not in quiz) 37.2.3.4 Machine production within specification</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p><p>The probability that a machine will produce all bolts in a production run within specification is {@p@}. A sample of {@n@} machines is taken at random. Calculate the probability that</p><ol class="HELM_parts">

<li><p>all {@n@} machines will produce all bolts within specification [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p></li>

<li><p>{@n-1@} or {@n@} machines will produce all bolts within specification [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p></li>

<li><p>at least {@n-2@} machines will produce all bolts within specification [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p></li>

</ol><p><b>Note:</b> please round your answer up to \(4\) decimal places, if requred.</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p><b>SOLUTION:</b></p><p>As we suspect, this experiment follows a Binomial distribution&nbsp;\(\text{P} (X=r)=\&nbsp; ^n\!C_rp^r(1-p)^{n-r}\) where \(p\) is the probability of single `success' which is the number of machines that produce all bolts within specification</p>
<ol class="HELM_parts">
  <li>\(\text{P} (X={@n@})=\  ^{{@n@}}\!C_{@n@}({@p@})^{@n@}({@1-p@})^0=({@p@})^{@n@}={@ta_1@}\)</li>
  <li><p><span class="math display">\(\begin{aligned}
\text{P} [(X={@n-1@})\cup (X={@n@})]&amp;=\text{P} (X={@n-1@})+\text{P} (X={@n@})\\
&amp;= \ ^{{@n@}}\!C_{@n-1@}({@p@})^{@n-1@}({@1-p@}) + {@ta_1@} \\ &amp;=&nbsp;({@n@})\times({@p@})^{@n-1@}({@1-p@}) + {@ta_1@}\\
&amp;={@ta_2@}\end{aligned}\)</span></p></li>
  <li>\(\begin{aligned} \text{P} (X\ge {@n-2@})=\text{P} [(X={@n-2@})\cup (X={@n-1@})\cup (X={@n@})] &amp;= \ ^{{@n@}}\!C_{@n-2@}({@p@})^{@n-2@}({@1-p@})^2 + {@ta_2@} \\ &amp;= {@ta_3@} \end{aligned}\)</li></ol>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2020070100</text>
    </stackversion>
    <questionvariables>
      <text>p: 0.991 + 0.001*rand(7)
n: 5 + rand(4)
ta_1_prev: p^n
ta_1: decimalplaces(ta_1_prev,4)
ta_2_prev: n*(p^(n-1))*(1-p) + ta_1_prev
ta_2: decimalplaces(ta_2_prev,4)
ta_3: decimalplaces(0.5*n*(n-1)*(p^(n-2))*(1-p)^2 + ta_2_prev,4)</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
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    <questionnote format="moodle_auto_format">
      <text>\(\text{P} (X={@n@})= {@ta_1@} \), \( \text{P} [(X={@n-1@})\cup (X={@n@})] = {@ta_2@} \), \( \text{P} (X\ge {@n-2@})= {@ta_3@} \)</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
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    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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