// Numbas version: exam_results_page_options {"name": "Calculate probabilities from binomial distribution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"thisnumber": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(number1<6,random(2..3), if(number1<8,random(2..4),random(3..6)))", "description": "", "name": "thisnumber"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "descx1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"number of chocolate chip cookies in our sample:\"", "description": "", "name": "descx1"}, "thismany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(15..20)", "description": "", "name": "thismany"}, "post": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"% of biscuits made by a baker are chocolate chip cookies.\"", "description": "", "name": "post"}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tprob1,3)", "description": "", "name": "prob1"}, "this": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"our selection contains exactly \"", "description": "", "name": "this"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(thatnumber=1,0,1)", "description": "", "name": "v"}, "prob": {"templateType": "anything", "group": "Ungrouped variables", "definition": "thismany/100", "description": "", "name": "prob"}, "tprob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(number1,thisnumber)*prob^thisnumber*(1-prob)^(number1-thisnumber)", "description": "", "name": "tprob1"}, 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"thisnumber", "what", "things", "descx", "tol", "prob", "thisaswell", "else", "thismany", "number1", "post", "prob2", "prob1", "thatnumber", "this", "v", "tprob1", "tprob2", "sd"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Calculate probabilities from binomial distribution", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "number1", "minValue": "number1", "correctAnswerFraction": false, "marks": 0.25, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prob", "minValue": "prob", "correctAnswerFraction": false, "marks": 0.25, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "number1*thismany/100", "minValue": 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Assuming a binomial distribution for $X$ , {descX}, write down the values of $n$ and $p$.
\n$X \\sim \\operatorname{bin}(n,p)$
\n$n=\\; $?[[0]] $p=\\;$?[[1]]
\nFind $\\operatorname{E}[X]$ the expected {descX1}
\n$\\operatorname{E}[X]=$?[[2]]
\nFind the standard deviation for the {descX1}
\nStandard deviation = ? [[3]] (to 3 decimal places).
\n\n \n \n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\nFind the probability that {this} $\\var{thisnumber}$ {things}
\n$\\operatorname{P}(X=\\var{thisnumber})=$? [[0]] (to 3 decimal places).
\n\n
Find the probability that {thisaswell} {thatnumber} {things}
\nProbability = ? [[1]] (to 3 decimal places).
\n\n \n \n", "showCorrectAnswer": true, "marks": 0}], "statement": "\n{pre} $\\var{thismany}$ {post}
\n{something} $\\var{number1}$ {else}
\n\n\n \n \n", "tags": ["acc1012", "ACC1012", "binomial distribution", "Binomial Distribution", "Binomial distribution", "checked2015", "expectation", "expected number", "probabilities", "probability", "Probability", "sc", "standard deviation", "statistical distributions", "statistics"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t \t\t \t\t \t\t \t\t
31/12/2012:
\n\t\t \t\t \t\t \t\t \t\tCan be configured to other applications using the string variables supplied. Hence added tag sc.
\n\t\t \t\t \t\t \t\t \t\tNot as yet properly tested.
\n\t\t \t\t \t\t \t\t \n\t\t \t\t \t\t \n\t\t \t\t \n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "\n\t\t \t\t \t\t \t\tApplication of the binomial distribution given probabilities of success of an event.
\n\t\t \t\t \t\t \t\tFinding probabilities using the binomial distribution.
\n\t\t \t\t \t\t \n\t\t \t\t \n\t\t \n\t\t"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\na)
\n1. $X \\sim \\operatorname{bin}(\\var{number1},\\var{prob})$, so $n= \\var{number1},\\;\\;p=\\var{prob}$.
\n2. The expectation is given by $\\operatorname{E}[X]=n\\times p=\\var{number1}\\times \\var{prob}=\\var{number1*prob}$
\n3. $\\operatorname{stdev}(X)=\\sqrt{n\\times p \\times (1-p)}=\\sqrt{\\var{number1}\\times \\var{prob} \\times \\var{1-prob}}=\\var{sd}$ to 3 decimal places.
\nb)
\n1. \\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\dbinom{\\var{number1}}{\\var{thisnumber}}\\times\\var{prob}^{\\var{thisnumber}}\\times(1-\\var{prob})^{\\var{number1-thisnumber}}\\\\& =& \\var{comb(number1,thisnumber)} \\times\\var{prob}^{\\var{thisnumber}}\\times\\var{1-prob}^{\\var{number1-thisnumber}}\\\\&=&\\var{prob1}\\end{eqnarray*} \\] to 3 decimal places.
\n\n
2.
\n\\[ \\begin{eqnarray*}\\operatorname{P}(X \\leq \\var{thatnumber})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\& =& \\simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})}\\\\& =& \\var{prob2}\\end{eqnarray*} \\]
\nto 3 decimal places.
\n\n\n \n \n", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}