// Numbas version: finer_feedback_settings
{"name": "IP3.3", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"scale": {"definition": "map(b*x+c,x,a)", "type": "list", "language": "jme", "parameters": [["a", "list"], ["b", "number"], ["c", "number"]]}}, "ungrouped_variables": ["a", "b", "d", "p", "pmf", "q", "twice", "x8", "expx", "t", "m2", "x2", "x3", "x0", "x1", "x6", "x7", "x4", "x5"], "name": "IP3.3", "tags": ["PMF", "Probability", "choose without replacement", "expectation", "expected value", "mass function", "pmf", "probabilities", "probability", "probability mass function", "random choice", "statistics", "tested1", "udf", "without replacement"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>First we find the sums that can occur by simply adding two different numbers together from the given.</p>\n<p>Note that these are pairs of different numbers as we choose without replacement.</p>\n<p>We get:</p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td align=\"left\"><strong>$X=x$</strong></td>\n<td>$\\var{pmf[0]}$</td>\n<td>$\\var{pmf[1]}$</td>\n<td>$\\var{pmf[2]}$</td>\n<td>$\\var{pmf[3]}$</td>\n<td>$\\var{pmf[4]}$</td>\n<td>$\\var{pmf[5]}$</td>\n<td>$\\var{pmf[6]}$</td>\n<td>$\\var{pmf[7]}$</td>\n<td>$\\var{pmf[8]}$</td>\n</tr>\n</tbody>\n</table>\n<p>We have to find the probabilities that each of these sums occur.</p>\n<p>There are $10$ ways of selecting $2$ numbers from the $5$ given, but it may be the case that two different pairs produce the same sum.</p>\n<p>In this case we find that there are two ways of producing the sum $\\var{twice}$. All other sums have only one way.</p>\n<p>So since each selection of a pair of numbers has probability $0.1$.</p>\n<p>The probability of producing the sum $\\var{twice}$ is $0.2$ , and the other sums have probability $0.1$ .</p>\n<h4>b)</h4>\n<p>The expectation is given by:</p>\n<p>\\[ \\begin{eqnarray*} E[X]&amp;=&amp; \\sum xP(X=x)\\\\ &amp;=&amp;\\simplify[]{ {pmf[0]}*{x0}+{pmf[1]}*{x1}+{pmf[2]}*{x2}+{pmf[3]}*{x3}+{pmf[4]}*{x4}+{pmf[5]}*{x5}+{pmf[6]}*{x6}+{pmf[7]}*{x7}+{pmf[8]}*{x8}}\\\\ &amp;=&amp;\\var{expx} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <p>Input the possible values of $X$ in the following table.</p>\n      \n      \n      \n      <p>You have to input  these in increasing order.</p>\n      \n      \n      \n      <p>Also find  the probability mass function $P(X=x)$ values and input them into the table.</p>\n      \n      \n      \n      <p>Input all values as exact values.</p>\n      \n      \n      \n      <table border=\"2\">\n      \n      <tr><td align=\"left\"><b>$X=x$</b></td><th>[[0]]</th><th>[[1]]</th><th>[[2]]</th><th>[[3]]</th><th>[[4]]</th><th>[[5]]</th><th>[[6]]</th><th>[[7]]</th><th>[[8]]</th></tr>\n      \n      <tr><td align=\"left\"><b>$P(X=x)$</b></td><td>[[9]]</td><td>[[10]]</td><td>[[11]]</td><td>[[12]]</td><td>[[13]]</td><td>[[14]]</td><td>[[15]]</td><td>[[16]]</td><td>[[17]]</td></tr>\n      \n      </table>\n      \n      \n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[0]}", "minValue": "{pmf[0]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[1]}", "minValue": "{pmf[1]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[2]}", "minValue": "{pmf[2]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[3]}", "minValue": "{pmf[3]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[4]}", "minValue": "{pmf[4]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[5]}", "minValue": "{pmf[5]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[6]}", "minValue": "{pmf[6]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[7]}", "minValue": "{pmf[7]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{pmf[8]}", "minValue": "{pmf[8]}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x0}", "minValue": "{x0}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x1}", "minValue": "{x1}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x2}", "minValue": "{x2}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x3}", "minValue": "{x3}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x4}", "minValue": "{x4}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x5}", "minValue": "{x5}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x6}", "minValue": "{x6}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x7}", "minValue": "{x7}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.1, "maxValue": "{x8}", "minValue": "{x8}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>Find the expectation, $E[X]$ of $X$.</p>\n      <p>Input as an exact decimal.</p>\n      <p>$E[X]=\\;\\;$[[0]]</p>\n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1.2, "maxValue": "{expx}", "minValue": "{expx}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n    <p>Two of the integers $\\var{d[0]},\\;\\var{d[1]}, \\;\\var{d[2]}, \\;\\var{d[3]}, \\;\\var{d[4]} $ are chosen at random, without replacement.</p>\n  <p>Let $X$ denote the sum of the chosen two values.</p>\n  <p>Answer the following two parts:</p>\n    ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "[[0,1,2,3,7],[0,1,2,4,6],[0,1,3,5,7]]", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "[[1,2,3,4,5,7,8,9,10],[1,2,3,4,5,6,7,8,10],[1,3,4,5,6,7,8,10,12]]", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "scale(a[t],p,q)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "twice": {"definition": "switch(t=0,pmf[2],t=1,pmf[5],pmf[6])", "templateType": "anything", "group": "Ungrouped variables", "name": "twice", "description": ""}, "x8": {"definition": "0.1", "templateType": "anything", "group": "Ungrouped variables", "name": "x8", "description": ""}, "q": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "p": {"definition": "random(1,2,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "pmf": {"definition": "scale(b[t],p,2*q)", "templateType": "anything", "group": "Ungrouped variables", "name": "pmf", "description": ""}, "expx": {"definition": "x0*pmf[0]+x1*pmf[1]+x2*pmf[2]+x3*pmf[3]+x4*pmf[4]+x5*pmf[5]+x6*pmf[6]+x7*pmf[7]+x8*pmf[8]", "templateType": "anything", "group": "Ungrouped variables", "name": "expx", "description": ""}, "t": {"definition": "random(0..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "m2": {"definition": "if(t=0,0.2,0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "m2", "description": ""}, "x2": {"definition": "if(t=0,0.2,0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "x2", "description": ""}, "x3": {"definition": "0.1", "templateType": "anything", "group": "Ungrouped variables", "name": "x3", "description": ""}, "x0": {"definition": "0.1", "templateType": "anything", "group": "Ungrouped variables", "name": "x0", "description": ""}, "x1": {"definition": "0.1", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}, "x6": {"definition": "if(t=2,0.2,0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "x6", "description": ""}, "x7": {"definition": "0.1", "templateType": "anything", "group": "Ungrouped variables", "name": "x7", "description": ""}, "x4": {"definition": "0.1", "templateType": "anything", "group": "Ungrouped variables", "name": "x4", "description": ""}, "x5": {"definition": "if(t=1,0.2,0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "x5", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>7/07/2012:</strong></p>\n        \t\t        <p>Added tags.</p>\n        \t\t        <p>Checked calculations.</p>\n        \t\t        <p><strong>22/07/2012:</strong></p>\n        \t\t        <p>Added description.</p>\n        \t\t        <p>Improved display, in particular replaced $E(X)$ by $E[X]$.</p>\n        \t\t        <p>Corrected typo in Advice (\"if\" instead of \"of\").</p>\n        \t\t        <p>Ticked stats extension box.</p>\n        \t\t        <p><strong>31/07/2012:</strong></p>\n        \t\t        <p>Added tags.</p>\n        \t\t        <p>Question appears to be working correctly.</p>\n        \t\t        <p><strong>20/12/2012:</strong></p>\n        \t\t        <p>Improved display of integers in statement.</p>\n        \t\t        <p>Added tag udf for user-defined functions.</p>\n        \t\t        <p>One such:</p>\n        \t\t        <p>scale(a,b,c)=map(b*x+c,x,a), scaling the list a.</p>\n        \t\t        <p>Calculation OK. Added tested1 tag.</p>\n        \t\t", "description": "<p>Two numbers from a set of $5$ numbers are chosen at random, without replacement. Find the distribution $X$ of their&nbsp;sum and $E[X]$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}