// Numbas version: exam_results_page_options
{"name": "Poisson_3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "prompt": "<p>Calculate, correct to three decimal places, the probability that at least&nbsp;\\(\\var{n}\\) goals will be scored in a particular match. &nbsp; &nbsp;[[0]]</p>", "gaps": [{"variableReplacements": [], "showPrecisionHint": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "precisionPartialCredit": 0, "mustBeReduced": false, "minValue": "{p}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "precisionType": "dp", "mustBeReducedPC": 0, "strictPrecision": false, "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{p}", "precision": "3", "allowFractions": false, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 1}], "showCorrectAnswer": true, "scripts": {}}], "rulesets": {}, "advice": "<p>The Poisson distribution \\(P(X=k)=\\frac{e^{-\\lambda}\\lambda^k}{k!}\\)</p>\n<p>In this problem \\(\\lambda=\\var{m}\\)</p>\n<p>P(at least \\(\\var{n}\\) goals) = \\(P(X\\ge\\var{n})\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X\\ge\\var{n})=1-P(X&lt;\\var{n})\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X\\ge\\var{n})=1-\\left((P(X=0)+P(X=1)+...P(X=\\simplify{{n}-1})\\right)\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X=0)=\\var{p_0}\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X=1)=\\var{p_1}\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X=2)=\\var{p_2}\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X=3)=\\var{p_3}\\)</p>\n<p style=\"padding-left: 90px;\">\\(P(X=4)=\\var{p_4}\\)</p>\n<p style=\"padding-left: 90px;\"></p>\n<p style=\"padding-left: 90px;\">\\(P(X\\ge\\var{n})=\\var{p}\\)</p>", "variable_groups": [], "variables": {"p_1": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "p_1", "definition": "e^(-{m})*({m})"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "p", "definition": "if(n<5,1-({p_0}+{p_1}+{p_2}+{p_3}),1-({p_0}+{p_1}+{p_2}+{p_3}+{p_4}))"}, "m": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "name": "m", "definition": "random(1..2.2#0.1)"}, "p_0": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "p_0", "definition": "e^(-{m})"}, "p_4": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "p_4", "definition": "(e^(-{m})*{m}^4)/24"}, "p_3": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "p_3", "definition": "(e^(-{m})*{m}^3)/6"}, "n": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "name": "n", "definition": "random(4..5#1)"}, "p_2": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "p_2", "definition": "(e^(-{m})*{m}^2)/2\n"}}, "name": "Poisson_3", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "extensions": [], "functions": {}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["m", "n", "p", "p_0", "p_1", "p_2", "p_3", "p_4"], "statement": "<p>The long term arithmetic mean&nbsp;number of goals scored in a premiership soccer match&nbsp;is \\(\\var{m}\\)&nbsp;&nbsp;</p>\n<p></p>\n<p></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/"}]}