// Numbas version: finer_feedback_settings
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Find,&nbsp;</p>\n<p>(i) $P(X&gt;\\var{r4})$</p>\n<p>Answer:[[3]]</p>\n<p>(ii) $P(X\\leq\\var{r5})$</p>\n<p>Answer:[[4]]</p>\n<p>(iii) $P(X&lt;\\var{r6})$</p>\n<p>Answer:[[5]]</p>\n<p>(iv) $P(X\\geq\\var{r7})$</p>\n<p>Answer:[[6]]</p>", "unitTests": [], "marks": 0, "customName": "Question 1", "gaps": [{"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans1}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans1}", "marks": 1, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "4", "correctAnswerFraction": false, 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is placed in every {q}<sup>th</sup>&nbsp;packet of crisps.&nbsp;Let X be the r.v. &ldquo;No. of packets bought to get a gift&rdquo;. Calculate the probability of first getting a gift after buying,</p>\n<p>(i) exactly {n1} packets.</p>\n<p>Answer:&nbsp;[[0]]</p>\n<p>(ii) at most {n2} packets.</p>\n<p>Answer:&nbsp;[[1]]</p>\n<p>(iii) at least&nbsp;{n3} packets.</p>\n<p>Answer:[[2]]</p>\n<p>(iv) at most&nbsp;{n4} packets.&nbsp;</p>\n<p>Answer:[[3]]</p>\n<p></p>\n<p>(b) 25% of new drivers have passed their driving test 1st time. Let X be the r.v. \"No. of tests until a student passes\". Calculate the probability of a student passing,&nbsp;</p>\n<p>(i) on their {n5}th attempt.&nbsp;</p>\n<p>Answer:[[4]]</p>\n<p>(ii) in no more than {n6} attempts.&nbsp;</p>\n<p>Answer:[[5]]</p>\n<p>(iii) in at least {n7} attempts.&nbsp;</p>\n<p>Answer:[[6]]</p>", "unitTests": [], "marks": 0, "customName": "Question 2", "gaps": [{"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans8}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans8}", "marks": 1, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "4", 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"showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans10}", "marks": "2", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "4", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0}, {"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans11}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans11}", "marks": "2", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "4", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0}, {"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans12}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans12}", "marks": 1, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "4", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0}, {"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans13}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans13}", "marks": 1, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "2", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0}, {"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": 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true}, {"prompt": "<p>(a) Two fair dice are thrown repeatedly until a double six appears. Let X be the r.v. \"No. of rolls before the double six appeared\". Find,</p>\n<p>(i) $P(X &ge; \\var{n8})$</p>\n<p>Answer:[[0]]</p>\n<p>(ii) $E(X)$</p>\n<p>Answer:[[1]]</p>", "unitTests": [], "marks": 0, "customName": "Question 3", "gaps": [{"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans15}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans15}", "marks": "2", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "4", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0}, {"mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{Ans16}", "customName": "", "showPrecisionHint": true, "scripts": {}, "showFeedbackIcon": true, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "useCustomName": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "adaptiveMarkingPenalty": 0, "minValue": "{Ans16}", "marks": 1, "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "precision": "0", "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0}], "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "useCustomName": true, "sortAnswers": false, "variableReplacements": [], "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true}], "extensions": ["stats"], "variable_groups": [{"name": "Question 1", "variables": ["p1", "r1", "Ans1", "r2", "Ans2", "r3", "Ans3", "p2", "r4", "Ans4", "r5", "Ans5", "r6", "Ans6", "r7", "j6", "Ans7", "j1"]}, {"name": "Question 2", "variables": ["q", "p3", "n1", "Ans8", "n2", "Ans9", "n3", "j3", "Ans10", "n4", "Ans11", "q2", "p4", "n5", "Ans12", "n6", "Ans13", "n7", "Ans14", "j4"]}, {"name": "Question 3", "variables": ["f1", "p5", "n8", "j5", "Ans15", "Ans16"]}], "variables": {"p1": {"templateType": "randrange", "definition": "random(0.01 .. 0.5#0.01)", "name": "p1", "group": "Question 1", "description": ""}, "p2": {"templateType": "randrange", "definition": "random(0.01 .. 0.5#0.01)", "name": "p2", "group": "Question 1", "description": ""}, "r3": {"templateType": "randrange", "definition": "random(8 .. 10#1)", "name": "r3", "group": "Question 1", "description": ""}, "p4": {"templateType": "anything", "definition": "q2/100", "name": "p4", "group": "Question 2", "description": ""}, "r7": {"templateType": "number", "definition": "5", "name": "r7", "group": "Question 1", "description": ""}, "r6": {"templateType": "number", "definition": "4", "name": "r6", "group": "Question 1", "description": ""}, "f1": {"templateType": "number", "definition": "36", "name": "f1", "group": "Question 3", "description": ""}, "Ans1": {"templateType": "anything", "definition": "geometricpdf(r1,p1)", "name": "Ans1", "group": "Question 1", "description": ""}, "Ans14": {"templateType": "anything", "definition": "1-geometriccdf(n7-1,p4)", "name": "Ans14", "group": "Question 2", "description": ""}, "Ans3": {"templateType": "anything", "definition": "geometricpdf(r3,p1)", "name": "Ans3", "group": "Question 1", "description": ""}, "q2": {"templateType": "anything", "definition": "25", "name": "q2", "group": "Question 2", "description": ""}, "Ans15": {"templateType": "anything", "definition": "1-geometriccdf(n8-1,p5)", "name": "Ans15", "group": "Question 3", "description": ""}, "Ans7": {"templateType": "anything", "definition": "1-geometriccdf(r7-1,p2)", "name": "Ans7", "group": "Question 1", "description": ""}, "Ans12": {"templateType": "anything", "definition": "geometricpdf(n5,p4)", "name": "Ans12", "group": "Question 2", "description": ""}, "r2": {"templateType": "randrange", "definition": "random(5 .. 7#1)", "name": "r2", "group": "Question 1", "description": ""}, "Ans9": {"templateType": "anything", "definition": "geometriccdf(n2,p3)", "name": "Ans9", "group": "Question 2", "description": ""}, "Ans2": {"templateType": "anything", "definition": "geometricpdf(r2,p1)", "name": "Ans2", "group": "Question 1", "description": ""}, "p5": {"templateType": "anything", "definition": "1/f1", "name": "p5", "group": "Question 3", "description": ""}, "j5": {"templateType": "anything", "definition": "n8-1", "name": "j5", "group": "Question 3", "description": "<p>j5</p>"}, "Ans10": {"templateType": "anything", "definition": "1-geometriccdf(n3-1,p3)", "name": "Ans10", "group": "Question 2", "description": ""}, "n4": {"templateType": "randrange", "definition": "random(8 .. 10#1)", "name": "n4", "group": "Question 2", "description": ""}, "Ans6": {"templateType": "anything", "definition": "geometriccdf(r6-1,p2)", "name": "Ans6", "group": "Question 1", "description": ""}, "n3": {"templateType": "randrange", "definition": "random(6 .. 7#1)", "name": "n3", "group": "Question 2", "description": ""}, "Ans5": {"templateType": "anything", "definition": "geometriccdf(r5,p2)", "name": "Ans5", "group": "Question 1", "description": ""}, "n1": {"templateType": "randrange", "definition": "random(4 .. 5#1)", "name": "n1", "group": "Question 2", "description": ""}, "j6": {"templateType": "anything", "definition": "r7-1", "name": "j6", "group": "Question 1", "description": ""}, "n7": {"templateType": "randrange", "definition": "random(3 .. 4#1)", "name": "n7", "group": "Question 2", "description": ""}, "j4": {"templateType": "anything", "definition": "n7-1", "name": "j4", "group": "Question 2", "description": ""}, "Ans4": {"templateType": "anything", "definition": "1-geometriccdf(r4,p2)", "name": "Ans4", "group": "Question 1", "description": ""}, "j3": {"templateType": "anything", "definition": "n3-1", "name": "j3", "group": "Question 2", "description": ""}, "r1": {"templateType": "randrange", "definition": "random(2 .. 4#1)", "name": "r1", "group": "Question 1", "description": ""}, "p3": {"templateType": "anything", "definition": "1/q", "name": "p3", "group": "Question 2", "description": ""}, "Ans13": {"templateType": "anything", "definition": "geometriccdf(n6,p4)", "name": "Ans13", "group": "Question 2", "description": ""}, "n5": {"templateType": "randrange", "definition": "random(4 .. 5#1)", "name": "n5", "group": "Question 2", "description": ""}, "r5": {"templateType": "number", "definition": "3", "name": "r5", "group": "Question 1", "description": ""}, "Ans8": {"templateType": "anything", "definition": "geometricpdf(n1,p3)", "name": "Ans8", "group": "Question 2", "description": ""}, "n2": {"templateType": "randrange", "definition": "random(2 .. 3#1)", "name": "n2", "group": "Question 2", "description": ""}, "Ans16": {"templateType": "anything", "definition": "1/p5", "name": "Ans16", "group": "Question 3", "description": ""}, "n8": {"templateType": "randrange", "definition": "random(10 .. 15#1)", "name": "n8", "group": "Question 3", "description": ""}, "Ans11": {"templateType": "anything", "definition": "geometriccdf(n4,p3)", "name": "Ans11", "group": "Question 2", "description": ""}, "q": {"templateType": "randrange", "definition": "random(4 .. 6#1)", "name": "q", "group": "Question 2", "description": ""}, "j1": {"templateType": "anything", "definition": "r6-1", "name": "j1", "group": "Question 1", "description": ""}, "n6": {"templateType": "randrange", "definition": "random(6 .. 7#1)", "name": "n6", "group": "Question 2", "description": ""}, "r4": {"templateType": "number", "definition": "2", "name": "r4", "group": "Question 1", "description": ""}}, "statement": "<p>This quiz tests the calculations for Geometric probabilities.&nbsp; Before you start you may wish to note down the probability mass function and cumulative density function formulae to help answer the questions.</p>", "functions": {}, "name": "Geometric Distribution", "metadata": {"description": "", "licence": "None specified"}, "advice": "<p>Here are some tips on how to set out your work, as well as how to complete the questions. None of the answers given below have been rounded to what the question states.&nbsp;</p>\n<p><strong>Question 1</strong></p>\n<p>1a: $X\\sim Geo(\\var{P1})$</p>\n<p>(i)&nbsp;$P(X= x)=p(1-p)^{x-1} \\rightarrow P(X=\\var{r1}) = \\var{p1}(1-\\var{p1})^{\\var{r1}-1} = \\var{Ans1}$</p>\n<p>(ii)&nbsp;$P(X= x)=p(1-p)^{x-1} \\rightarrow P(X=\\var{r2}) =&nbsp; \\var{p1}(1-\\var{p1})^{\\var{r2}-1} = \\var{Ans2}$</p>\n<p>(iii)&nbsp;$P(X= x)=p(1-p)^{x-1} \\rightarrow P(X=\\var{r3}) = \\var{p1}(1-\\var{p1})^{\\var{r3}-1} = \\var{Ans3}$</p>\n<p><strong>&nbsp;</strong></p>\n<p>1b:&nbsp;$X\\sim Geo(\\var{P2})$</p>\n<p>(i)&nbsp;$P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X&gt;\\var{r4}) = 1-P(X\\leq\\var{r4}) = 1-(1-(1-\\var{p2})^{\\var{r4}})=\\var{Ans4}$</p>\n<p>(ii) $P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\leq\\var{r5}) =&nbsp; 1-(1-\\var{p2})^{\\var{r5}}=\\var{Ans5}$</p>\n<p>(iii)&nbsp;$P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X&lt;\\var{r6}) = P(X\\leq \\var{r6}-1)= P(X\\leq \\var{j1})= 1-(1-\\var{p2})^\\var{j1}=\\var{Ans6}$</p>\n<p>(iv)&nbsp;$P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\geq\\var{r7}) =1-P(X\\leq \\var{j6}) =1-(1-(1-p)^\\var{j6})=\\var{Ans7}$</p>\n<p></p>\n<p><strong>Question 2</strong></p>\n<p>2a: $ X\\sim Geo(\\var{P3})$</p>\n<p>(i) $P(X= x)=p(1-p)^{x-1} \\rightarrow P(X=\\var{n1}) = \\frac{1}{\\var{q}}(1-\\frac{1}{\\var{q}})^{\\var{n1}-1} = \\var{Ans8}$</p>\n<p>(ii) $P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X=\\var{n2})= 1-(1-\\frac{1}{\\var{q}})^{\\var{n2}-1} = \\var{Ans9}$</p>\n<p>(iii)&nbsp;$P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\geq \\var{n3}) = 1-P(X\\leq \\var{n3}-1) = 1-P(X\\leq\\var{j3}) = 1-(1-(1-p)^\\var{j3})=\\var{Ans10}$</p>\n<p>(iv)&nbsp;$P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\leq \\var{n4}) = 1-(1-p)^\\var{n4}=\\var{Ans11}$</p>\n<p></p>\n<p>2b:&nbsp;$ X\\sim Geo(\\var{P4})$</p>\n<p>(i) $P(X= x)=p(1-p)^{x-1} \\rightarrow P(X=\\var{n5}) = \\var{p4}(1-\\var{p4})^{\\var{n5}-1} = \\var{Ans12}$</p>\n<p>(ii) $P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\leq \\var{n6}) = 1-(1-\\var{p4})^{\\var{n6}} = \\var{Ans13}$</p>\n<p>(iii) $P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\geq \\var{n7}) = 1-P(X\\leq\\var{n7}-1)=1-P(X\\leq \\var{j4})=1-(1-(1-\\var{p4})^\\var{j4}) = \\var{Ans14}$</p>\n<p></p>\n<p><strong>Question 3</strong></p>\n<p>3a: $X\\sim Geo(\\var{p5})$</p>\n<p>(i) $P(X\\leq x)=1-(1-p)^{\\mid x\\mid} \\rightarrow P(X\\geq \\var{n8}) = 1-P(X\\leq\\var{n8}-1)=1-P(X\\leq \\var{j5})=1-(1-(1-\\var{p5})^{\\var{j5}}) = \\var{Ans15}$</p>\n<p>(ii) $E(X)=\\frac{1}{p} =\\frac{1}{\\frac{1}{\\var{f1}}} = \\var{Ans16}$</p>\n<p></p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": [], "type": "question", "contributors": [{"name": "Ellen Marshall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2533/"}, {"name": "Brad Allison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3394/"}]}]}], "contributors": [{"name": "Ellen Marshall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2533/"}, {"name": "Brad Allison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3394/"}]}