// Numbas version: finer_feedback_settings
{"name": "Calculate probabilities using the exponential distribution, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.5..0.8#0.01)", "description": "", "name": "r"}, "thismany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..10#0.2)", "description": "", "name": "thismany"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5#0.1)", "description": "", "name": "m"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(e^(-thismany/m1),3)", "description": "", "name": "p1"}, "place": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\n      'London',\n      'Manhattan',\n      'Mexico City',\n      'Beijing',\n      'Los Angeles',\n      'Buenos Aires',\n      'Bangkok'\n      )", "description": "", "name": "place"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(e^(-thismany/m),3)", "description": "", "name": "p"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(r*m,1)", "description": "", "name": "m1"}, "stuff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\n      'carbon monoxide',\n      'Freon-22',\n      'hydrogen sulphide',\n      'mercury',\n      'polynuclear aromatic hydrocarbon',\n      'crystalline silica')", "description": "", "name": "stuff"}}, "ungrouped_variables": ["p1", "thismany", "m", "p", "stuff", "m1", "tol", "place", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Calculate probabilities using the exponential distribution, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "prompt": "<p>Find the probability that the {stuff} concentration exceeds $\\var{thismany}$ parts per million in a one hour period.</p>\n      <p>Input your answer to 3 decimal places.</p>", "minValue": "p-tol", "maxValue": "p+tol", "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "prompt": "<p>A control strategy reduced the mean to $\\var{m1}$ parts per million.</p>\n      <p>Now find the probability that a concentration exceeds $\\var{thismany}$ parts per million in a one hour period.</p>\n      <p>Input your answer to 3 decimal places.</p>", "minValue": "p1-tol", "maxValue": "p1+tol", "marks": 1, "showPrecisionHint": false}], "statement": "<p>One hour {stuff} concentrations in samples of air taken at a location in {place} have an approximate exponential distribution with mean $\\var{m}$ parts per million.</p>", "tags": ["checked2015", "continuous distributions", "distributions", "exponential distribution", "MAS1604", "MAS2304", "Probability", "statistical distributions", "statistics"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/01/2013:</strong></p>\n    <p><strong><br /></strong>First draft completed.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculating simple probabilities using the exponential distribution.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The random variable $X$ is {stuff} concentration {ppm} and $ \\displaystyle X \\sim \\operatorname{Exp}\\left(\\frac{1}{\\var{m}}\\right )$.</p>\n  <p>Hence the probability that $X \\lt x$ is $P(X \\lt x)=1-e^{-x/\\var{m}}$.</p>\n  <p>a)</p>\n  <p>$P(X \\gt \\var{thismany})=1-P(X \\lt \\var{thismany})=1-(1-e^{-\\var{thismany}/\\var{m}})=\\var{p}$ to 3 decimal places.</p>\n  <p>&nbsp;</p>\n  <p>b) Changing the mean value gives:</p>\n  <p>$P(X \\gt \\var{thismany})=1-P(X \\lt \\var{thismany})=1-(1-e^{-\\var{thismany}/\\var{m1}})=\\var{p1}$ to 3 decimal places.</p>\n  <p>&nbsp;</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}