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{"name": "Calculating expected values given a table of probabilities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Calculating expected values given a table of probabilities", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question assesses&nbsp;the students ability to find the expected number of times an event&nbsp;occurs given the probability of the event occurring for a single trial and the total number of trials.</p>"}, "statement": "<p>The were 542 reported injuries which caused footballers to miss either games or days of training during the 2017 Premier League season.</p>\n<p>The table below shows the probability that an injured player suffered a particular injury, denoted $P(\\text{Injury})$.</p>\n<table width=\"437\" height=\"248\" style=\"height: 237px;\" border=\"-\">\n<tbody>\n<tr style=\"height: 37px;\">\n<td style=\"width: 126.33px; height: 37px; text-align: center; background-color: #c8c5c4;\">Injury</td>\n<td style=\"width: 145.37px; height: 37px; text-align: center; background-color: #c8c5c4;\">$P(\\text{Injury})$</td>\n<td style=\"width: 116.3px; height: 37px; text-align: center; background-color: #c8c5c4;\">Structure</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126.33px; height: 38px; text-align: center;\">Hamstring</td>\n<td style=\"width: 145.37px; height: 38px; text-align: center;\">$\\var{Hamstring}$</td>\n<td style=\"width: 116.3px; height: 38px; text-align: center;\">Muscle</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126.33px; height: 38px; text-align: center;\">Knee</td>\n<td style=\"width: 145.37px; height: 38px; text-align: center;\">$\\var{Knee}$</td>\n<td style=\"width: 116.3px; height: 38px; text-align: center;\">Joint</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126.33px; height: 38px; text-align: center;\">Ankle</td>\n<td style=\"width: 145.37px; height: 38px; text-align: center;\">$\\var{Ankle}$</td>\n<td style=\"width: 116.3px; height: 38px; text-align: center;\">Joint</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126.33px; height: 38px; text-align: center;\">Illness</td>\n<td style=\"width: 145.37px; height: 38px; text-align: center;\">$\\var{Illness}$</td>\n<td style=\"width: 116.3px; height: 38px; text-align: center;\">Immune System</td>\n</tr>\n</tbody>\n</table>\n<p>A random sample of $\\var{no_people}$ of these injured or ill athletes attended your clinic.</p>", "rulesets": {}, "advice": "<p></p>\n<p>\\[\\text{Expected number of times&nbsp;an event occurs} = \\text{Probability of event} \\times \\text{Number of trials}.\\]&nbsp;</p>\n<p></p>\n<h4>a)</h4>\n<p>The expected number of people attending the clinic who have had a knee injury is</p>\n<p>\\[\\text{probabilty of having a knee injury} \\times \\text{number of people attending clinic} =&nbsp;<br/>\\var{Knee} \\times \\var{no_people} = \\var{{Knee}*{no_people}}.<br/>\\]</p>\n<h4>b)</h4>\n<p>Let us denote the probability that a person has damaged a joint $P(\\text{Joint})$. We can see that there are two injuries which belong to the Joint structure:&nbsp;Knee and Ankle. Therefore, it is true that</p>\n<p></p>\n<p>\\[<br/>\\begin{align}<br/>P(\\text{Joint}) &amp;= P(\\text{Knee})+P(\\text{Ankle})\\\\<br/>&amp;= \\var{Knee}+\\var{Ankle}\\\\<br/>&amp;= \\var{Knee+Ankle}.<br/>\\end{align}<br/>\\]</p>\n<p>Then the expected number of people who have damaged a joint is</p>\n<p>\\[<br/>\\var{Knee + Ankle} \\times \\var{no_people} = \\var{dpformat(({Knee + Ankle})*{no_people},2)},<br/>\\]</p>\n<p>which rounds to $\\var{dpformat(({Knee + Ankle})*{no_people},0)}$ to the nearest integer.</p>", "variable_groups": [], "functions": {}, "extensions": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["Hamstring", "Knee", "Ankle", "Illness", "no_people"], "parts": [{"showCorrectAnswer": true, "marks": 1, "type": "numberentry", "mustBeReduced": false, "prompt": "<p>How many athletes&nbsp;would you expect to have had a knee injury?</p>", "correctAnswerStyle": "plain", "scripts": {}, "extendBaseMarkingAlgorithm": true, "minValue": "{no_people}*{Knee}", "unitTests": [], "maxValue": "{no_people}*{Knee}", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"showCorrectAnswer": true, "precisionPartialCredit": 0, "strictPrecision": true, "prompt": "<p>How many athletes would you have expected to damaged a joint?</p>", "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "maxValue": "{no_people}*({Knee}+{Ankle})", "type": "numberentry", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showPrecisionHint": true, "mustBeReduced": false, "precisionType": "dp", "scripts": {}, "extendBaseMarkingAlgorithm": true, "minValue": "{no_people}*({Knee}+{Ankle})", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerFraction": false, "precision": 0, "customMarkingAlgorithm": ""}], "variables": {"Hamstring": {"group": "Ungrouped variables", "definition": "random(0.2..0.31 #0.05)", "description": "", "templateType": "anything", "name": "Hamstring"}, "no_people": {"group": "Ungrouped variables", "definition": "random(100..180 #20)", "description": "<p>Number of people who see a movie.</p>", "templateType": "anything", "name": "no_people"}, "Ankle": {"group": "Ungrouped variables", "definition": "(1-(Hamstring+Knee))*3/5", "description": "", "templateType": "anything", "name": "Ankle"}, "Illness": {"group": "Ungrouped variables", "definition": "1-(Hamstring+Knee+Ankle)", "description": "", "templateType": "anything", "name": "Illness"}, "Knee": {"group": "Ungrouped variables", "definition": "random(0.4..0.51 #0.05)", "description": "", "templateType": "anything", "name": "Knee"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}