// Numbas version: exam_results_page_options
{"name": "Find probabilities from 2D frequency table,", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "statement": "\n    <p>A survey was conducted to obtain information on {this}. A random sample of {things}s gave :</p>\n    <p>&nbsp;</p>\n    <table>\n    <tbody>\n    <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>{Cats}</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>{At[0]}</strong></td>\n  <td style=\"vertical-align: middle;border-bottom: 1px solid;\"><strong>{At[1]}</strong></td>\n    <td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid;\"><strong>{At[2]}</strong></td>\n    <td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid;\"><strong>Total</strong></td>\n    </tr>\n    <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>{cat[0]}</strong></td>\n    <td>{r[0][0]}</td>\n    <td>{r[0][1]}</td>\n    <td>{r[0][2]}</td>\n    <td style=\"vertical-align: middle;border-left: 1px solid; border-right: 1px solid;\">{sumr[0]}</td>\n    </tr>\n    <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>{cat[1]}</strong></td>\n    <td>{r[1][0]}</td>\n    <td>{r[1][1]}</td>\n    <td>{r[1][2]}</td>\n    <td style=\"vertical-align: middle; border-left: 1px solid;border-right: 1px solid;\">{sumr[1]}</td>\n    </tr>\n    <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>{cat[2]}</strong></td>\n    <td>{r[2][0]}</td>\n    <td>{r[2][1]}</td>\n    <td>{r[2][2]}</td>\n    <td style=\"vertical-align: middle; border-left: 1px solid;border-right: 1px solid;\">{sumr[2]}</td>\n    </tr>\n    <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>{cat[3]}</strong></td>\n    <td>{r[3][0]}</td>\n    <td>{r[3][1]}</td>\n    <td>{r[3][2]}</td>\n    <td style=\"vertical-align: middle; border-left: 1px solid;border-right: 1px solid;\">{sumr[3]}</td>\n    </tr>\n    <tr>\n    <td style=\"vertical-align: middle; border-right: 1px solid; border-bottom: 1px solid;border-top: 1px solid;\"><strong>Totals</strong></td>\n    <td style=\"vertical-align: middle; border-bottom: 1px solid;border-top: 1px solid;\">{tc[0]}</td>\n    <td style=\"vertical-align: middle; border-bottom: 1px solid;border-top: 1px solid;\">{tc[1]}</td>\n    <td style=\"vertical-align: middle; border-bottom: 1px solid;border-top: 1px solid;\">{tc[2]}</td>\n    <td style=\"vertical-align: middle; border-left: 1px solid;border-right: 1px solid; border-bottom: 1px solid;border-top: 1px solid;\">{tot}</td>\n    </tr>\n    </tbody>\n    </table>\n    \n    ", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Finding probabilities from a survey giving a table of data on the alcohol consumption of males. This can be easily adapted to data from other types of surveys.</p>", "notes": "\n        \t\t<p><strong>29/12/2012:</strong></p>\n        \t\t    <p>Added tags and description.</p>\n        \t\t    <p>This is easily configurable to other surveys by changing the variables used for labelling. Added the tag sc to denote this. Also added tag, table, as there is a table included.</p>\n        \t\t    <p>The column labels in the table need to be centred.</p>\n        \t\t    <p>The presentation and layout of the questions should be improved and made consistent.</p>\n        \t\t    <p>There is a user-defined function norm(a,x,n) which takes a numeric list a and changes the entry a[x] so that the sum of entries in a is n. Added the tag udf.&nbsp;</p>\n        \t\t    <p>Calculations correct on testing.</p>\n        \t\t"}, "showQuestionGroupNames": false, "tags": ["ACE2013", "checked2015", "MAS1403"], "ungrouped_variables": ["somecat", "ve", "othercats", "at", "ce1", "ans", "cat", "drk1", "things", "u1", "tot", "catattrib2", "cats", "tc", "w1", "oneof", "tc2", "tc3", "tc1", "catattrib1", "a", "drkpair", "this", "n", "we2", "drk", "r", "u", "t", "w", "v", "sumr", "ce2"], "parts": [{"gaps": [{"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[0]", "showPrecisionHint": false, "minValue": "ans[0]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[1]", "showPrecisionHint": false, "minValue": "ans[1]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[2]", "showPrecisionHint": false, "minValue": "ans[2]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[3]", "showPrecisionHint": false, "minValue": "ans[3]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[4]", "showPrecisionHint": false, "minValue": "ans[4]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[5]", "showPrecisionHint": false, "minValue": "ans[5]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}], "marks": 0, "type": "gapfill", "prompt": "\n        <p>Find the following probabilities that a randomly chosen {things} involved in this survey:(Enter all probabilities to 3 decimal places).</p>\n      <p>1) is {somecat}: Probability =? [[0]]</p>\n      <p>2) is {drk}: &nbsp;Probability = ? [[1]]</p>\n      <p>3) is either {oneof}: Probability =? [[2]]</p>\n      <p>4) {drkpair}: Probability =? [[3]]</p>\n      <p>5) {catattrib1}: Probability =? [[4]]</p>\n      <p>6) {catattrib2}: Probability=? [[5]]</p>\n        ", "showCorrectAnswer": true, "scripts": {}}, {"gaps": [{"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[6]", "showPrecisionHint": false, "minValue": "ans[6]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[7]", "showPrecisionHint": false, "minValue": "ans[7]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}], "marks": 0, "type": "gapfill", "prompt": "\n        <p>Find the probability (to 3 decimal places) that two randomly selected {things}s in this survey are</p>\n      <p>7) both {somecat}: &nbsp;Probability = ? [[0]]</p>\n      <p>8) both {drk1}: Probability =? [[1]]</p>\n        ", "showCorrectAnswer": true, "scripts": {}}, {"gaps": [{"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[8]", "showPrecisionHint": false, "minValue": "ans[8]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}, {"allowFractions": false, "marks": 1, "type": "numberentry", "maxValue": "ans[9]", "showPrecisionHint": false, "minValue": "ans[9]", "showCorrectAnswer": true, "correctAnswerFraction": false, "scripts": {}}], "marks": 0, "type": "gapfill", "prompt": "\n        <p>Given that a randomly selected {things} in this survey is {drk}, what is the probability that he:</p>\n      <p>9) is {somecat}: Probability = ? [[0]]</p>\n      <p>10) is {othercats}: Probability =? [[1]]</p>\n      <p>&nbsp;</p>\n        ", "showCorrectAnswer": true, "scripts": {}}], "functions": {"norm": {"language": "javascript", "definition": "\n        var b=a;\n        var s=-b[x];\n        for(i=0;i<b.length;i++)\n      {s+=b[i];}\n        b[x]=n-s;\n        return b;\n        /* adjusts the xth entry so that the sum over entries is n*/\n        \n        ", "type": "list", "parameters": [["a", "list"], ["x", "number"], ["n", "number"]]}}, "preamble": {"css": "", "js": ""}, "variables": {"tc1": {"group": "Ungrouped variables", "description": "", "definition": "round(n/random(2.5..3.5#0.1))", "name": "tc1", "templateType": "anything"}, "u1": {"group": "Ungrouped variables", "description": "", "definition": "random(0..2)", "name": "u1", "templateType": "anything"}, "n": {"group": "Ungrouped variables", "description": "", "definition": "random(1200..2300#2)", "name": "n", "templateType": "anything"}, "w1": {"group": "Ungrouped variables", "description": "", "definition": "random(1..6)", "name": "w1", "templateType": "anything"}, "v": {"group": "Ungrouped variables", "description": "", "definition": "random(1..6)", "name": "v", "templateType": "anything"}, "things": {"group": "Ungrouped variables", "description": "", "definition": "\"male\"", "name": "things", "templateType": "anything"}, "drkpair": {"group": "Ungrouped variables", "description": "", "definition": "switch(u=0,\"drinks alcohol\",u=1,At[0]+' or ' + At[2],At[0]+\" or \" +At[1])", "name": "drkpair", "templateType": "anything"}, "tc3": {"group": "Ungrouped variables", "description": "", "definition": "round(n/random(6.5..7.5#0.1))", "name": "tc3", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "[[round(tc1/random(7.5..8.5#0.1)),0,round(tc1/random(6.5..7.5#0.1)),round(tc1/random(19.5..20.5#0.1))],[round(tc2/random(3.5..4.5#0.1)),0,round(tc2/random(19.5..20.5#0.1)),round(tc2/random(19.5..20.5#0.1))],[round(tc3/random(2.5..3.5#0.1)),0,random(5..15),round(tc3/random(14.5..15.5#0.1))]]", "name": "a", "templateType": "anything"}, "catattrib2": {"group": "Ungrouped variables", "description": "", "definition": "switch(w1=1,cat[2]+\" and \"+At[0],w1=2,cat[2]+\" and \"+At[1],w1=3,cat[2]+\" and \"+At[2],w1=4,cat[3]+\" and \"+At[0],w1=5,cat[3]+\" and \"+At[1],cat[3]+\" and \"+At[2])", "name": "catattrib2", "templateType": "anything"}, "we2": {"group": "Ungrouped variables", "description": "", "definition": "if(u=0, if(t=0,r[1][0]+r[2][0], if(t=1,r[0][0]+r[3][0], if(t=2,r[0][0]+r[1][0],r[0][0]+r[2][0]))),if(u=1, if(t=0,r[1][1]+r[2][1],  if(t=1,r[0][1]+r[3][1], if(t=2,r[0][1]+r[1][1],r[0][1]+r[2][1]))),if(t=0,r[1][2]+r[2][2],  if(t=1,r[0][2]+r[3][2], if(t=2,r[0][2]+r[1][2], r[0][2]+r[2][2])))))", "name": "we2", "templateType": "anything"}, "r": {"group": "Ungrouped variables", "description": "", "definition": "transpose(matrix(map(norm(a[y],1,tc[y]),y,0..2)))", "name": "r", "templateType": "anything"}, "ce2": {"group": "Ungrouped variables", "description": "", "definition": "switch(w1=1, r[2][0],w1=2, r[2][1],w1=3, r[2][2],w1=4,r[3][0],w1=5, r[3][1],r[3][2])", "name": "ce2", "templateType": "anything"}, "u": {"group": "Ungrouped variables", "description": "", "definition": "random(0..2)", "name": "u", "templateType": "anything"}, "t": {"group": "Ungrouped variables", "description": "", "definition": "random(0..3)", "name": "t", "templateType": "anything"}, "somecat": {"group": "Ungrouped variables", "description": "", "definition": "cat[t]", "name": "somecat", "templateType": "anything"}, "at": {"group": "Ungrouped variables", "description": "", "definition": "[\"teetotal\",\"drinking 1-20 units/week\",\"drinking 21+ units/week\"]", "name": "at", "templateType": "anything"}, "tc2": {"group": "Ungrouped variables", "description": "", "definition": "n-tc1-tc3", "name": "tc2", "templateType": "anything"}, "this": {"group": "Ungrouped variables", "description": "", "definition": "\"alcohol consumption\"", "name": "this", "templateType": "anything"}, "catattrib1": {"group": "Ungrouped variables", "description": "", "definition": "switch(w=1,cat[0]+\" and \"+At[0],w=2,cat[0]+\" and \"+At[1],w=3,cat[0]+\" and \"+At[2],w=4,cat[1]+\" and \"+At[0],w=5,cat[1]+\" and \"+At[1],cat[1]+\" and \"+At[2])", "name": "catattrib1", "templateType": "anything"}, "ce1": {"group": "Ungrouped variables", "description": "", "definition": "switch(w=1, r[0][0],w=2, r[0][1],w=3, r[0][2],w=4,r[1][0],w=5, r[1][1],r[1][2])", "name": "ce1", "templateType": "anything"}, "drk1": {"group": "Ungrouped variables", "description": "", "definition": "At[u1]", "name": "drk1", "templateType": "anything"}, "othercats": {"group": "Ungrouped variables", "description": "", "definition": "switch(t=0,cat[1]+\" or \"+ cat[2] ,t=1,cat[0]+\" or \"+cat[3],t=2,cat[0]+\" or \"+cat[1],t=3,cat[0]+\" or \"+cat[2],cat[0]+\" or \"+cat[2])", "name": "othercats", "templateType": "anything"}, "tot": {"group": "Ungrouped variables", "description": "", "definition": "sum(tc)", "name": "tot", "templateType": "anything"}, "ve": {"group": "Ungrouped variables", "description": "", "definition": "switch(v=1,sumr[0]+ sumr[1] ,v=2,sumr[0]+sumr[2],v=3,sumr[0]+sumr[3],v=4,sumr[1]+sumr[2],v=5,sumr[1]+sumr[3],sumr[2]+sumr[3])", "name": "ve", "templateType": "anything"}, "cat": {"group": "Ungrouped variables", "description": "", "definition": "['single','married','divorced','widowed']", "name": "cat", "templateType": "anything"}, "w": {"group": "Ungrouped variables", "description": "", "definition": "random(1..6)", "name": "w", "templateType": "anything"}, "tc": {"group": "Ungrouped variables", "description": "", "definition": "[tc1,tc2,tc3]", "name": "tc", "templateType": "anything"}, "sumr": {"group": "Ungrouped variables", "description": "", "definition": "map(sum(list(r[y])),y,0..3)", "name": "sumr", "templateType": "anything"}, "ans": {"group": "Ungrouped variables", "description": "", "definition": "map(precround(x,3),x,[sumr[t]/n,tc[u]/n,ve/n,1-tc[u]/n,ce1/n,ce2/n,sumr[t]*(sumr[t]-1)/(n*(n-1)),tc[u1]*(tc[u1]-1)/(n*(n-1)),r[t][u]/tc[u],we2/tc[u]])", "name": "ans", "templateType": "anything"}, "drk": {"group": "Ungrouped variables", "description": "", "definition": "At[u]", "name": "drk", "templateType": "anything"}, "oneof": {"group": "Ungrouped variables", "description": "", "definition": "switch(v=1,cat[0]+\" or \"+ cat[1] ,v=2,cat[0]+\" or \"+cat[2],v=3,cat[0]+\" or \"+cat[3],v=4,cat[1]+\" or \"+cat[2],v=5,cat[1]+\" or \"+cat[3],cat[2]+\" or \"+cat[3])", "name": "oneof", "templateType": "anything"}, "cats": {"group": "Ungrouped variables", "description": "", "definition": "\"Marital Status\"", "name": "cats", "templateType": "anything"}}, "question_groups": [{"pickQuestions": 0, "questions": [], "name": "", "pickingStrategy": "all-ordered"}], "type": "question", "advice": "\n    <p>1) The total number of {somecat} {things}s is $\\var{sumr[t]}$ hence the probability that a random {things} from this survey is {somecat} is $\\displaystyle \\frac{ \\var{sumr[t]}}{\\var{n}}=\\var{ans[0]}$ to 3 decimal places.</p>\n  <p>&nbsp;</p>\n  <p>2) The total number of &nbsp;{things}s who are {drk} is&nbsp;$\\var{tc[u]}$&nbsp;hence the probability that a random {things} from this survey is {drk} is&nbsp;$\\displaystyle \\frac{ \\var{tc[u]}}{\\var{n}}=\\var{ans[1]}$&nbsp;&nbsp;to 3 decimal places.</p>\n  <p>&nbsp;</p>\n  <p>3) Looking at the table there are $\\var{ve}$ {things}s that are {oneof}. Hence the probability is&nbsp;$\\displaystyle \\frac{ \\var{ve}}{\\var{n}}=\\var{ans[2]}$&nbsp;&nbsp;to 3 decimal places.</p>\n  <p>&nbsp;</p>\n  <p>4) These are the {things}s that are not {drk}, and hence there are $\\var{n}-\\var{tc[u]}=\\var{n-tc[u]}$ of them (see answer to part b)), and the probability of randomly selecting one is&nbsp;&nbsp;$\\displaystyle \\frac{ \\var{n-tc[u]}}{\\var{n}}=\\var{ans[3]}$&nbsp;to 3 decimal places.</p>\n  <p>&nbsp;</p>\n  <p>5)Looking at the table we see that the number corresponding to {catattrib1} is $\\var{ce1}$. Hence the probability of randomly selecting one is&nbsp;$\\displaystyle \\frac{ \\var{ce1}}{\\var{n}}=\\var{ans[4]}$&nbsp;to 3 decimal places.</p>\n  <p>6) As in the last question, looking at the table we see that the number corresponding to {catattrib2} is&nbsp;$\\var{ce2}$. Hence the probability of randomly selecting one is&nbsp;$\\displaystyle \\frac{ \\var{ce2}}{\\var{n}}=\\var{ans[5]}$&nbsp;to 3 decimal places.</p>\n  <p>&nbsp;</p>\n  <p>7) We know from question a) that the probability of selecting a {somecat} {things} is,&nbsp;&nbsp;$\\displaystyle \\frac{ \\var{sumr[t]}}{\\var{n}}$,&nbsp;after this we now have $\\var{sumr[t]-1} $ {somecat} {things}s amongst the $\\var{n-1}$ &nbsp;left, &nbsp;and the probability of yet again selecting one &nbsp;of these is&nbsp;$\\displaystyle \\frac{ \\var{sumr[t]-1}}{\\var{n-1}}$. So the probability of selecting two is &nbsp;$\\displaystyle \\frac{ \\var{sumr[t]}\\times&nbsp;&nbsp;\\var{sumr[t]-1}}{\\var{n}\\times\\var{n-1}}=\\var{ans[6]}$&nbsp;to 3 decimal places.</p>\n  <p>8) The probability of selecting a {things} who is {drk1} is&nbsp;$\\displaystyle \\frac{ \\var{tc[u1]}}{\\var{n}}$, after this we now have &nbsp;$\\var{tc[u1]-1}$ {drk1} {things}s amongst the $\\var{n-1}$ left, and&nbsp;the probability of yet again selecting one &nbsp;of these is&nbsp;$\\displaystyle \\frac{ \\var{tc[u1]-1}}{\\var{n-1}}$. So the probability of selecting two is &nbsp;$\\displaystyle \\frac{ \\var{tc[u1]}\\times&nbsp;&nbsp;\\var{tc[u1]-1}}{\\var{n}\\times\\var{n-1}}=\\var{ans[7]}$&nbsp;to 3 decimal places.</p>\n  <p>9) Since there are $\\var{r[t][u]}$ {somecat} {things}s from the &nbsp;$\\var{tc[u]}$&nbsp;{things}s that are {drk} the probability of selecting one &nbsp;is $\\displaystyle \\frac{\\var{r[t][u]}}{\\var{tc[u]}}= \\var{ans[8]}$ to 3 decimal places.</p>\n  <p>10) Since there are $\\var{we2}$ {othercats} {things}s from the $\\var{tc[u]}$ {things}s that are {drk} the probability of selecting one &nbsp;is&nbsp;$\\displaystyle \\frac{\\var{we2}}{\\var{tc[u]}}= \\var{ans[9]}$&nbsp;&nbsp; to 3 decimal places.</p>\n    ", "rulesets": {}, "name": "Find probabilities from 2D frequency table,", "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}