// Numbas version: exam_results_page_options
{"name": "Poisson Distribution (Dentist)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Poisson Distribution (Dentist)", "tags": [], "metadata": {"description": "<p>The number of patients arriving at a dentist&rsquo;s surgery each afternoon follows<br />a Poisson distribution, with a mean of four patients per hour.<br />Calculate the probability that in a particular one-hour period</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p><span>T</span><span>he number of patients arriving at a dentist&rsquo;s surgery each afternoon follows a Poisson distribution, with a mean of $\\var{l}$ patients per hour.<br/><br/></span></p>\n<p><span></span></p>\n<p>&nbsp;The Poisson distribution formula: $P(r)=\\frac{\\lambda^re^{-\\lambda}}{r!}$ or&nbsp;$P(r)=e^{-\\lambda}\\left[\\frac{\\lambda^r}{r!}\\right]$</p>\n<p></p>\n<p></p>\n<p><strong>Please give your answer to at least 3 decimal places.</strong></p>", "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><span>The probability that in a particular one hour period, the number of patients entering the waiting room</span><span>&nbsp;will be between&nbsp;$\\var{x}$ and $\\var{y}$ inclusive is given by:</span></p>\n<p>$P(\\var{x} \\leq X\\leq\\var{y}) = P(X=\\var{x}) + P(X=\\var{x+1}) +P(X=\\var{y})$</p>\n<p>where&nbsp;</p>\n<p>$P(X=\\var{x}) =\\frac{\\var{l}^{\\var{x}}e^{-\\var{l}}}{\\var{x}!}=\\var{prx}$</p>\n<p><span>$P(X=\\var{x+1}) =\\frac{\\var{l}^{\\var{x+1}}e^{-\\var{l}}}{\\var{x+1}!}=\\var{prx1}$</span></p>\n<p><span>$P(X=\\var{y}) =\\frac{\\var{l}^{\\var{y}}e^{-\\var{l}}}{\\var{y}!}=\\var{pry}$</span></p>\n<p>Hence&nbsp;</p>\n<p><span>$P(\\var{x} \\leq X \\leq \\var{y})=$ $\\var{prx}+\\var{prx1}+\\var{pry}=\\var{answer2}$</span></p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(10..50#5)", "description": "<p>time interval</p>", "templateType": "anything", "can_override": false}, "answer1": {"name": "answer1", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!", "description": "", "templateType": "anything", "can_override": false}, "prx": {"name": "prx", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!", "description": "", "templateType": "anything", "can_override": false}, "answer2": {"name": "answer2", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!+((e^-l)*(l^(x+1)))/(x+1)!+((e^-l)*(l^y))/y!", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "x+2", "description": "<p>upper value of X</p>", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..5#1)", "description": "<p>time hour</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(3..12)", "description": "<p>number of customers entering the shop</p>", "templateType": "anything", "can_override": false}, "prx1": {"name": "prx1", "group": "Ungrouped variables", "definition": "((e^-l)*(l^(x+1)))/(x+1)!", "description": "", "templateType": "anything", "can_override": false}, "pry": {"name": "pry", "group": "Ungrouped variables", "definition": "((e^-l)*(l^y))/y!", "description": "", "templateType": "anything", "can_override": false}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "random(5..10#1)", "description": "<p>average, lambda</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["l", "x", "n", "y", "t", "answer1", "answer2", "prx", "prx1", "pry"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write down the value of $\\lambda$ for one hour.</p>\n<p></p>\n<p>$\\lambda = $ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "l", "maxValue": "l", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><span>Calculate the probability that in a particular one-hour period exactly $\\var{x}$ patients will arrive. Please give your answer to at least 3 decimal places.<br/><br/><br/></span></p>\n<p>$P(r = \\var{x}) =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "minValue": "answer1-0.005", "maxValue": "answer1+0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "answer1-0.001", "maxValue": "answer1+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><span>Find the probability that in a particular one hour period, the number of patients entering the waiting room</span><span> will be between $\\var{x}$ and $\\var{y}$ inclusive. Please give your answer to at least 3 decimal places.</span></p>\n<p><span>&nbsp;</span></p>\n<p>$P(\\var{x}\\leq r \\leq\\var{y}) =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "minValue": "answer2 -0.005", "maxValue": "answer2 +0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "answer2 -0.005", "maxValue": "answer2 +0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}]}]}], "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}]}