// Numbas version: finer_feedback_settings
{"name": "BS3.3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["pre", "descx1", "something", "thisnumber", "what", "things", "descx", "tol", "prob", "thisaswell", "else", "thismany", "number1", "post", "prob2", "prob1", "thatnumber", "this", "v", "tprob1", "tprob2", "sd"], "name": "BS3.3", "tags": ["Binomial Distribution", "Binomial distribution", "Probability", "binomial distribution", "expectation", "expected number", "probabilities", "probability", "sc", "standard deviation", "statistical distributions", "statistics"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>1. $X \\sim \\operatorname{bin}(\\var{number1},\\var{prob})$, so $n= \\var{number1},\\;\\;p=\\var{prob}$.</p>\n<p>2. The expectation is given by $\\operatorname{E}[X]=n\\times p=\\var{number1}\\times \\var{prob}=\\var{number1*prob}$</p>\n<p>3. $\\operatorname{stdev}(X)=\\sqrt{n\\times p \\times (1-p)}=\\sqrt{\\var{number1}\\times \\var{prob} \\times \\var{1-prob}}=\\var{sd}$ to 3 decimal places.</p>\n<h4>b)</h4>\n<p>1. \\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &amp;=&amp; \\dbinom{\\var{number1}}{\\var{thisnumber}}\\times\\var{prob}^{\\var{thisnumber}}\\times(1-\\var{prob})^{\\var{number1-thisnumber}}\\\\&amp; =&amp; \\var{comb(number1,thisnumber)} \\times\\var{prob}^{\\var{thisnumber}}\\times\\var{1-prob}^{\\var{number1-thisnumber}}\\\\&amp;=&amp;\\var{prob1}\\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>2.&nbsp;</p>\n<p>\\[ \\begin{eqnarray*}\\operatorname{P}(X \\leq \\var{thatnumber})&amp; =&amp; \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\&amp; =&amp; \\simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})}\\\\&amp; =&amp; \\var{prob2}\\end{eqnarray*} \\]</p>\n<p>to 3 decimal places.</p>\n<p>&nbsp;</p>", "rulesets": {}, "parts": [{"prompt": "<p>Assuming a binomial distribution for $X$ , {descX}, write down the values of $n$ and $p$.</p>\n<p>$X \\sim \\operatorname{bin}(n,p)$</p>\n<p>$n=$&nbsp;[[0]] &nbsp; &nbsp; &nbsp; &nbsp;$p=$&nbsp;[[1]]</p>\n<p>Find $\\operatorname{E}[X]$ the expected {descX1}</p>\n<p>$\\operatorname{E}[X]=$&nbsp;[[2]]</p>\n<p>Find the standard deviation for the {descX1}</p>\n<p>Standard deviation = [[3]] (to 3 decimal places).</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.25, "maxValue": "number1", "minValue": "number1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.25, "maxValue": "prob", "minValue": "prob", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "number1*thismany/100", "minValue": "number1*thismany/100", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "sd+tol", "minValue": "sd-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Find the probability that {this} $\\var{thisnumber}$ {things}</p>\n<p>$\\operatorname{P}(X=\\var{thisnumber})=$&nbsp;[[0]] (to 3 decimal places).</p>\n<p>&nbsp;</p>\n<p>Find the probability that {thisaswell} {thatnumber} {things}</p>\n<p>Probability = [[1]] (to 3 decimal places).</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n  <p>{pre} $\\var{thismany}$ {post}</p>\n    <p>{something} $\\var{number1}$ {else}</p>\n    <p>&nbsp;</p>\n  \n    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"pre": {"definition": "' '", "templateType": "anything", "group": "Ungrouped variables", "name": "pre", "description": ""}, "descx1": {"definition": "\"number of chocolate chip cookies in our sample:\"", "templateType": "anything", "group": "Ungrouped variables", "name": "descx1", "description": ""}, "something": {"definition": "''", "templateType": "anything", "group": "Ungrouped variables", "name": "something", "description": ""}, "thisnumber": {"definition": "if(number1<6,random(2..3), if(number1<8,random(2..4),random(3..6)))", "templateType": "anything", "group": "Ungrouped variables", "name": "thisnumber", "description": ""}, "what": {"definition": "\"daily sales.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "what", "description": ""}, "things": {"definition": "\"chocolate chip cookies.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "things", "description": ""}, "descx": {"definition": "\"the number of chocolate chip cookies\"", "templateType": "anything", "group": "Ungrouped variables", "name": "descx", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "prob": {"definition": "thismany/100", "templateType": "anything", "group": "Ungrouped variables", "name": "prob", "description": ""}, "thisaswell": {"definition": "\"our selection contains no more than \"", "templateType": "anything", "group": "Ungrouped variables", "name": "thisaswell", "description": ""}, "else": {"definition": "\"biscuits are selected at random.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "else", "description": ""}, "thismany": {"definition": "random(15..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "thismany", "description": ""}, "number1": {"definition": "random(5..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "number1", "description": ""}, "post": {"definition": "\"% of biscuits made by a baker are chocolate chip cookies.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "post", "description": ""}, "prob2": {"definition": "precround(tprob2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob1": {"definition": "precround(tprob1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}, "thatnumber": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "thatnumber", "description": ""}, "this": {"definition": "\"our selection contains exactly \"", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "v": {"definition": "if(thatnumber=1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "tprob1": {"definition": "comb(number1,thisnumber)*prob^thisnumber*(1-prob)^(number1-thisnumber)", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob1", "description": ""}, "tprob2": {"definition": "if(thatnumber=2,(1-prob)^number1+number1*prob*(1-prob)^(number1-1)+number1*(number1-1)*prob^2*(1-prob)^(number1-2)/2,(1-prob)^number1+number1*prob*(1-prob)^(number1-1))", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob2", "description": ""}, "sd": {"definition": "precround(sqrt(number1*prob*(1-prob)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "sd", "description": ""}}, "metadata": {"notes": "\n        \t\t    \t\t            \t\t<p><strong>31/12/2012:</strong></p>\n        \t\t    \t\t            \t\t    <p>Can be configured to other applications using the string variables supplied. Hence added tag sc.</p>\n        \t\t    \t\t            \t\t    <p>Not as yet properly tested.</p>\n        \t\t    \t\t            \n        \t\t    \n        \t\t", "description": "\n        \t\t    \t\t<p>Application of the binomial distribution given probabilities of success of an event.</p>\n        \t\t    \t\t    <p>Finding probabilities using the binomial distribution.</p>\n        \t\t    \n        \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "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/"}]}