// Numbas version: exam_results_page_options
{"name": "binomial3", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"": {"parameters": [], "type": "number", "language": "jme", "definition": ""}}, "extensions": ["stats"], "advice": "<p>The probability of winning a prize&nbsp;in a particular raffle&nbsp;is \\(\\var{p1}\\).</p>\n<p>A person&nbsp;buys&nbsp;\\(\\var{n1}\\) raffle tickets&nbsp;</p>\n<p>The probability that the person will win at least \\(\\simplify{{n2}+1}\\) prizes can be expressed as \\(P(X\\ge \\simplify{{n2}+1})=1-P(X&lt;\\simplify{{n2}+1})\\)</p>\n<p></p>\n<p>The binomial distribution gives: \\(P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}\\)</p>\n<p></p>\n<p style=\"padding-left: 30px;\">\\(P(X=0)=\\binom{\\var{n1}}{0}(\\var{p1})^{0}(\\var{p2})^{\\var{n1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(P(X=0)=(1)(1)(\\simplify{{p2}^{{n1}}})\\)</p>\n<p style=\"padding-left: 30px;\">\\(P(X=0)=\\var{prob0}\\)</p>\n<p>Similarly</p>\n<p style=\"padding-left: 30px;\">\\(P(X=1)=\\binom{\\var{n1}}{1}(\\var{p1})^{1}(\\var{p2})^{\\simplify{{n1}-1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(P(X=1)=\\var{prob1}\\)</p>\n<p>And</p>\n<p style=\"padding-left: 30px;\">\\(P(X=2)=\\binom{\\var{n1}}{2}(\\var{p1})^{2}(\\var{p2})^{\\simplify{{n1}-2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(P(X=2)=\\var{prob2}\\)</p>\n<p>So</p>\n<p style=\"padding-left: 30px;\">\\(P(X\\ge 2)=1-(\\var{prob0}+\\var{prob1})=\\simplify{1-{prob0}-{prob1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(P(X\\ge 3)=1-\\var{prob0}+\\var{prob1}+\\var{prob2})=\\simplify{1-{prob0}-{prob1}-{prob2}}\\)</p>", "rulesets": {}, "name": "binomial3", "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(0.05..0.1#0.01)", "description": "", "templateType": "randrange"}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "if({n2}<2,1-(prob0+prob1),1-(prob0+prob1+prob2))", "description": "", "templateType": "anything"}, "prob1": {"name": "prob1", "group": "Ungrouped variables", "definition": "{n1}*{p1}*(1-{p1})^({n1}-1)", "description": "", "templateType": "anything"}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "normalcdf(2,0,1)", "description": "", "templateType": "anything"}, "prob2": {"name": "prob2", "group": "Ungrouped variables", "definition": "comb({n1},2)*{p1}^2*(1-{p1})^({n1}-2)", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(1..2#1)", "description": "", "templateType": "randrange"}, "prob0": {"name": "prob0", "group": "Ungrouped variables", "definition": "(1-{p1})^{n1}", "description": "", "templateType": "anything"}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "1-{p1}", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "1-{z}", "description": "", "templateType": "anything"}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(4..12#1)", "description": "", "templateType": "randrange"}}, "ungrouped_variables": ["p1", "n1", "n2", "prob0", "prob1", "prob2", "ans1", "p2", "z", "x"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "parts": [{"scripts": {}, "gaps": [{"strictPrecision": false, "mustBeReduced": false, "minValue": "ans1", "precisionPartialCredit": 0, "variableReplacements": [], "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showPrecisionHint": true, "type": "numberentry", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "showFeedbackIcon": true, "maxValue": "ans1", "showCorrectAnswer": true, "correctAnswerStyle": "plain"}], "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>If a person buys&nbsp;\\(\\var{n1}\\) raffle tickets calculate the probability that the person will win at least \\(\\simplify{{n2}+1}\\) prizes.&nbsp;[[0]]</p>\n<p></p>", "showCorrectAnswer": true, "marks": 0, "type": "gapfill"}], "preamble": {"js": "", "css": ""}, "tags": [], "statement": "<p>The probability of winning a prize&nbsp;in a particular raffle&nbsp;is \\(\\var{p1}\\).</p>", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}