// Numbas version: exam_results_page_options
{"name": "Simon's copy of Binomial Distribution (Cycling)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p><strong>Please give your answer to 3 decimal places.</strong></p>\n<p>It is estimated that $\\var{p_perc}$%&nbsp;of all CIT students cycle to college. A&nbsp;random sample of $\\var{n}$ CIT students is chosen.</p>\n<p></p>", "parts": [{"minValue": "(q^n)", "precision": "3", "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": true, "prompt": "<p>Calculate the probability that none of the <span>$\\var{n}$</span>&nbsp;students in the sample cycle to college.</p>", "scripts": {}, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "precisionPartialCredit": 0, "allowFractions": false, "precisionType": "dp", "mustBeReducedPC": 0, "type": "numberentry", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "maxValue": "(q^n)", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": "1"}, {"minValue": "answer2 ", "precision": "3", "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": true, "prompt": "<p>Calculate the probability that at least&nbsp;$\\var{r}$ of the&nbsp;<span>$\\var{n}$</span> students cycle to college.</p>", "scripts": {}, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "unitTests": [], "precisionPartialCredit": 0, "allowFractions": false, "precisionType": "dp", "mustBeReducedPC": 0, "type": "numberentry", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "maxValue": "answer2 ", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": "2"}], "variables": {"pr2": {"name": "pr2", "description": "<p>probability that r = 2</p>", "definition": "((n*(n-1))/2)*(p^2)*q^(n-2)", "group": "Ungrouped variables", "templateType": "anything"}, "p_perc": {"name": "p_perc", "description": "<p>percentage of students that cycle to college</p>", "definition": "p*100", "group": "Ungrouped variables", "templateType": "anything"}, "q": {"name": "q", "description": "<p>probability tha an individual does not cycle to college</p>", "definition": "1-p", "group": "Ungrouped variables", "templateType": "anything"}, "n": {"name": "n", "description": "<p>sample size</p>", "definition": "random(6..12)", "group": "Ungrouped variables", "templateType": "anything"}, "pr3": {"name": "pr3", "description": "<p>probability that r = 3</p>", "definition": "((n*(n-1)*(n-2))/6)*(p^3)*(q^(n-3))", "group": "Ungrouped variables", "templateType": "anything"}, "p": {"name": "p", "description": "<p>the probability that an individual student cycles to college</p>", "definition": "random(0.1..0.4#0.05)", "group": "Ungrouped variables", "templateType": "anything"}, "answer2": {"name": "answer2", "description": "", "definition": "1-answer1", "group": "Ungrouped variables", "templateType": "anything"}, "answer1": {"name": "answer1", "description": "", "definition": "if(r=2,pr0+pr1, pr0+pr1+pr2)", "group": "Ungrouped variables", "templateType": "anything"}, "qn": {"name": "qn", "description": "", "definition": "q^n", "group": "Ungrouped variables", "templateType": "anything"}, "pr0": {"name": "pr0", "description": "<p>probability that&nbsp;r = 0</p>", "definition": "q^n", "group": "Ungrouped variables", "templateType": "anything"}, "r": {"name": "r", "description": "<p>more than r of the students cycle to college</p>", "definition": "3", "group": "Ungrouped variables", "templateType": "anything"}, "r0": {"name": "r0", "description": "", "definition": "0", "group": "Ungrouped variables", "templateType": "anything"}, "n2": {"name": "n2", "description": "", "definition": "n-2", "group": "Ungrouped variables", "templateType": "anything"}, "pr1": {"name": "pr1", "description": "<p>probability that r = 1</p>", "definition": "n*p*q^(n-1)", "group": "Ungrouped variables", "templateType": "anything"}}, "tags": [], "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "<p>It is estimated that 30% of all CIT students cycle to college. If a random sample of eight CIT students is chosen, calculate the probability that...</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Simon's copy of Binomial Distribution (Cycling)", "ungrouped_variables": ["p", "p_perc", "n", "q", "r", "pr0", "pr1", "pr2", "pr3", "answer1", "answer2", "qn", "r0", "n2"], "variable_groups": [], "extensions": [], "advice": "<p><strong>(a)</strong></p>\n<p>If a&nbsp;random variable $X$&nbsp;follows a&nbsp;binomial distribution with parameters $n$&nbsp;and $p$, then&nbsp;the probability of $r$ successes out of $n$ trials is given by:</p>\n<p>$P(X=r)=\\binom{n}{r}p^{r}q^{n-r}$</p>\n<p>where</p>\n<p>$p=$ probability of success for each trial,&nbsp;$q=1-p=$probability of failure for each trial,&nbsp;and $\\binom{n}{r} = ^nC_r= \\frac{n!}{r!(n-r)!}$</p>\n<p></p>\n<p>The probability that a student cycles to college&nbsp;is $\\var{p}$, therefore $p=\\var{p}$ and $q=1-\\var{p}=\\var{q}$.</p>\n<p>We are interested in claculating the probability that none of the sample of $\\var{n}$ students cycle to college so $r=0$ and $n=\\var{n}$</p>\n<p>$P(X=\\var{r0})= \\binom{\\var{n}}{\\var{r0}}\\times \\var{p}^0 \\times \\var{q}^{\\var{n}-0}=\\var{precround(pr0,5)}$</p>\n<p><span></span></p>\n<p><strong>(b)</strong></p>\n<p><span>We are interested in claculating the probability that&nbsp;at least&nbsp;$\\var{r}$ of the&nbsp;<span>$\\var{n}$</span><span> students cycle to college. Let $X$ represent the number of students that cycle to college. We need to calculate:</span></span></p>\n<p><span>$P(X \\geq \\var{r}) = P(X= \\var{r}) + P(X= \\var{r+1})+...+ P(X=\\var{n})$</span></p>\n<p><span></span></p>\n<p>Since $P(X=\\var{r0})+P(X=\\var{r0+1})+...+P(X=\\var{n})=\\var{r0+1}$&nbsp;</p>\n<p><span>We may write&nbsp;</span></p>\n<p><span>$P(X \\geq \\var{r}) = 1-P(X= \\var{r0}) - P(X=\\var{r0+1})- P(X=\\var{r-1})$</span></p>\n<p><span></span></p>\n<p>where</p>\n<p><span>$P(X= \\var{r0})= \\binom{\\var{n}}{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}=\\var{precround(pr0,5)}$&nbsp;</span></p>\n<p><span>$P(X=1) = \\binom{\\var{n}}{\\var{1}}$ $\\var{p}^{1}$ $\\var{q}^{\\var{n}-1}$ $=\\var{precround(pr1,5)}$</span></p>\n<p><span>$P(X=2) =\\binom{\\var{n}}{\\var{2}}$&nbsp;</span>$\\var{p}^{2}$ $\\var{q}^{\\var{n}-2}$ $=\\var{precround(pr2,5)}$</p>\n<p></p>\n<p>Then&nbsp;</p>\n<p><span>$P(X \\geq \\var{r}) = 1-\\var{precround(qn,5)}-\\var{precround(pr1,5)}-\\var{precround(pr2,5)}=\\var{precround(answer2,5)}$</span></p>", "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}