// Numbas version: finer_feedback_settings
{"name": "Probability, expectation and standard deviation of binomial distribution", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "tags": ["checked2015", "MAS1403"], "variables": {"thisaswell": {"description": "", "templateType": "anything", "definition": "\"our selection contains no more than \"", "name": "thisaswell", "group": "Ungrouped variables"}, "prob1": {"description": "", "templateType": "anything", "definition": "precround(tprob1,3)", "name": "prob1", "group": "Ungrouped variables"}, "v": {"description": "", "templateType": "anything", "definition": "if(thatnumber=1,0,1)", "name": "v", "group": "Ungrouped variables"}, "descx": {"description": "", "templateType": "anything", "definition": "\"the number of chocolate chip cookies\"", "name": "descx", "group": "Ungrouped variables"}, "what": {"description": "", "templateType": "anything", "definition": "\"daily sales.\"", "name": "what", "group": "Ungrouped variables"}, "something": {"description": "", "templateType": "anything", "definition": "''", "name": "something", "group": "Ungrouped variables"}, "number1": {"description": "", "templateType": "anything", "definition": "random(5..12)", "name": "number1", "group": "Ungrouped variables"}, "sd": {"description": "", "templateType": "anything", "definition": "precround(sqrt(number1*prob*(1-prob)),3)", "name": "sd", "group": "Ungrouped variables"}, "prob": {"description": "", "templateType": "anything", "definition": "thismany/100", "name": "prob", "group": "Ungrouped variables"}, "prob2": {"description": "", "templateType": "anything", "definition": "precround(tprob2,3)", "name": "prob2", "group": "Ungrouped variables"}, "tprob1": {"description": "", "templateType": "anything", "definition": "binomialpdf(thisnumber,number1,prob)", "name": "tprob1", "group": "Ungrouped variables"}, "thismany": {"description": "", "templateType": "anything", "definition": "random(15..20)", "name": "thismany", "group": "Ungrouped variables"}, "tprob2": {"description": "", "templateType": "anything", "definition": "binomialcdf(thatnumber,number1,prob)", "name": "tprob2", "group": "Ungrouped variables"}, "pre": {"description": "", "templateType": "anything", "definition": "' '", "name": "pre", "group": "Ungrouped variables"}, "descx1": {"description": "", "templateType": "anything", "definition": "\"number of chocolate chip cookies in our sample:\"", "name": "descx1", "group": "Ungrouped variables"}, "thisnumber": {"description": "", "templateType": "anything", "definition": "if(number1<6,random(2..3), if(number1<8,random(2..4),random(3..6)))", "name": "thisnumber", "group": "Ungrouped variables"}, "else": {"description": "", "templateType": "anything", "definition": "\"biscuits are selected at random.\"", "name": "else", "group": "Ungrouped variables"}, "thatnumber": {"description": "", "templateType": "anything", "definition": "random(1,2)", "name": "thatnumber", "group": "Ungrouped variables"}, "things": {"description": "", "templateType": "anything", "definition": "\"chocolate chip cookies.\"", "name": "things", "group": "Ungrouped variables"}, "post": {"description": "", "templateType": "anything", "definition": "\"% of biscuits made by a baker are chocolate chip cookies.\"", "name": "post", "group": "Ungrouped variables"}, "this": {"description": "", "templateType": "anything", "definition": "\"our selection contains exactly \"", "name": "this", "group": "Ungrouped variables"}, "tol": {"description": "", "templateType": "anything", "definition": "0.001", "name": "tol", "group": "Ungrouped variables"}}, "name": "Probability, expectation and standard deviation of binomial distribution", "advice": "\n  <p>a)</p>\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    <p>b)</p>\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>&nbsp;</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>\n  ", "showQuestionGroupNames": false, "statement": "<p>{pre} $\\var{thismany}$ {post}</p>\n    <p>{something} $\\var{number1}$ {else}</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>", "variable_groups": [], "metadata": {"description": "\n            \t\t<p>Application of the binomial distribution given probabilities of success of an event.</p>\n            \t\t    <p>Finding probabilities using the binomial distribution.</p>\n            \t\t", "licence": "Creative Commons Attribution 4.0 International", "notes": "<p><strong>31/12/2012:</strong></p>\n        <p>Can be configured to other applications using the string variables supplied. Hence added tag sc.</p>\n        <p>Not as yet properly tested.</p>\n        <p><strong>13/01/2013:</strong></p>\n        <p>Used stats extension functions binomialpdf and binomialcdf instead of calculating insitu.</p>"}, "parts": [{"gaps": [{"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "marks": 0.25, "minValue": "number1", "maxValue": "number1", "scripts": {}, "correctAnswerFraction": false}, {"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "marks": 0.25, "minValue": "prob", "maxValue": "prob", "scripts": {}, "correctAnswerFraction": false}, {"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "marks": 0.5, "minValue": "number1*thismany/100", "maxValue": "number1*thismany/100", "scripts": {}, "correctAnswerFraction": false}, {"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "marks": 1, "minValue": "sd-tol", "maxValue": "sd+tol", "scripts": {}, "correctAnswerFraction": false}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}, "prompt": "\n      <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=\\; $?[[0]] &nbsp; &nbsp; &nbsp; &nbsp;$p=\\;$?[[1]]</p>\n        <p>Find $\\operatorname{E}[X]$ the expected {descX1}</p>\n        <p>$\\operatorname{E}[X]=$?[[2]]</p>\n        <p>Find the standard deviation for the {descX1}</p>\n        <p>Standard deviation = ? [[3]] (to 3 decimal places).</p>\n      "}, {"gaps": [{"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "marks": 1, "minValue": "prob1-tol", "maxValue": "prob1+tol", "scripts": {}, "correctAnswerFraction": false}, {"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false, "marks": 1, "minValue": "prob2-tol", "maxValue": "prob2+tol", "scripts": {}, "correctAnswerFraction": false}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}, "prompt": "\n      <p>Find the probability that {this} $\\var{thisnumber}$ {things}</p>\n        <p>$\\operatorname{P}(X=\\var{thisnumber})=$? [[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>\n      "}], "ungrouped_variables": ["pre", "descx1", "something", "thisnumber", "what", "things", "descx", "tol", "prob", "thisaswell", "else", "thismany", "number1", "post", "prob2", "prob1", "thatnumber", "this", "v", "tprob1", "tprob2", "sd"], "type": "question", "preamble": {"js": "", "css": ""}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0, "name": ""}], "functions": {}, "rulesets": {}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}