// Numbas version: exam_results_page_options
{"name": "Simon's copy of Clodagh's copy of Poisson Distribution (printing errors)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"type": "question", "statement": "<p><strong>Please give your answer to at least 3 decimal places.</strong></p>\n<p><span>Printing errors in the work produced by a particular film occur randomly at an average rate of $\\var{l}$ per page.</span></p>", "variable_groups": [], "variables": {"y": {"name": "y", "templateType": "anything", "definition": "3", "group": "Ungrouped variables", "description": "<p>upper value of X</p>"}, "pr2": {"name": "pr2", "templateType": "anything", "definition": "((e^-l2)*(l2^2))/2!", "group": "Ungrouped variables", "description": ""}, "x": {"name": "x", "templateType": "anything", "definition": "random(0..2)", "group": "Ungrouped variables", "description": "<p>number of customers entering the shop</p>"}, "l2": {"name": "l2", "templateType": "anything", "definition": "l*n", "group": "Ungrouped variables", "description": ""}, "l": {"name": "l", "templateType": "anything", "definition": "random(0.05..0.4#0.1)", "group": "Ungrouped variables", "description": "<p>average, lambda</p>"}, "pr0": {"name": "pr0", "templateType": "anything", "definition": "((e^-l2)*(l2^0))/0!", "group": "Ungrouped variables", "description": ""}, "pr1": {"name": "pr1", "templateType": "anything", "definition": "((e^-l2)*(l2^1))/1!", "group": "Ungrouped variables", "description": ""}, "answer2": {"name": "answer2", "templateType": "anything", "definition": "if(y=3, ((e^-l2)*(l2^0))/0!+((e^-l2)*(l2^(1)))/(1)!+((e^-l2)*(l2^2))/2!,((e^-l2)*(l2^0))/0!+((e^-l2)*(l2^(1)))/(1)!)", "group": "Ungrouped variables", "description": ""}, "answer1": {"name": "answer1", "templateType": "anything", "definition": "((e^-l)*(l^x))/x!", "group": "Ungrouped variables", "description": ""}, "n": {"name": "n", "templateType": "anything", "definition": "random(6..15#1)", "group": "Ungrouped variables", "description": "<p>time interval</p>"}}, "name": "Simon's copy of Clodagh's copy of Poisson Distribution (printing errors)", "preamble": {"css": "", "js": ""}, "rulesets": {}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "<p>rebelmaths</p>\n<p>Printing errors in the work produced by a particular film occur randomly at an average rate of p&nbsp;per page.</p>\n<p><br /> i.What is the probability that a one page document will contain x1&nbsp;printing error(s)?<br /><br /> ii.If a n&nbsp;page document is printed, calculate the probability of having more than x2&nbsp;errors. Assume a Poisson distribution.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": ["rebelmaths"], "ungrouped_variables": ["l", "x", "n", "y", "answer1", "answer2", "l2", "pr0", "pr1", "pr2"], "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "parts": [{"minValue": "answer1-0.001", "marks": "3", "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacements": [], "correctAnswerFraction": false, "maxValue": "answer1+0.001", "showPrecisionHint": false, "prompt": "<p><span>What is the probability that a one page document will contain exactly $\\var{x}$&nbsp;printing error(s)?</span></p>", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"minValue": "answer2 -0.001", "marks": "5", "allowFractions": false, "scripts": {}, "type": "numberentry", "variableReplacements": [], "correctAnswerFraction": false, "maxValue": "answer2 +0.001", "showPrecisionHint": false, "prompt": "<p><span>If a $\\var{n}$ page document is printed, calculate the probability of having less&nbsp;than $\\var{y}$&nbsp;errors.&nbsp;</span></p>", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "extensions": [], "advice": "<p><strong>Part (a)</strong></p>\n<p>Remember that for a Poisson random variable:<br/>\\begin{align}<br/>\\operatorname{P}(X=x)&amp;=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\<br/>\\end{align}</p>\n<p>1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{x}) &amp;=&amp; \\frac{\\var{l} ^ {\\var{x}}e ^ { -\\var{l}}} {\\var{x}!}\\\\&amp; =&amp; \\var{answer1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>&nbsp;</p>\n<p><strong>Part (b)</strong></p>\n<p>2.&nbsp;For a $\\var{n}$ page document, $\\lambda=\\var{n} \\times \\var{l} = \\var{l2}$</p>\n<p><span>The probability of having less&nbsp;than $\\var{y}$&nbsp;errors is given by:</span></p>\n<p>$P(X &lt; \\var{y}) = P(X=0) + P(X=1) +P(X=2)$</p>\n<p>where&nbsp;</p>\n<p>$P(X=0) =\\frac{\\var{l2}^{0}e^{-\\var{l2}}}{0!}=\\var{pr0}$</p>\n<p><span>$P(X=1) =\\frac{\\var{l2}^{1}e^{-\\var{l2}}}{1!}=\\var{pr1}$</span></p>\n<p><span>$P(X=1) =\\frac{\\var{l2}^{2}e^{-\\var{l2}}}{2!}=\\var{pr2}$</span></p>\n<p>Hence&nbsp;</p>\n<p><span>$P(X &lt; \\var{y})$ = $\\var{pr0}+\\var{pr1}+\\var{pr2}=\\var{answer2}$</span></p>", "showQuestionGroupNames": false, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}