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{"name": "Simon's copy of 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": [{"variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "advice": "<p>If we are given the probability of an&nbsp;event occurring&nbsp;in a single&nbsp;trial&nbsp;then we can calculate the expected number of times that this&nbsp;event would&nbsp;occur&nbsp;in a larger number of trials.</p>\n<p>To do this, we multiply the probability of&nbsp;the event occurring in a single trial by the&nbsp;total number of trials:</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>We are given the probabilities that someone buys&nbsp;a ticket to see each film in the table below.</p>\n<table width=\"436\" height=\"248\" border=\"-\" style=\"height: 248px;\">\n<tbody>\n<tr style=\"height: 37px;\">\n<td style=\"width: 126px; height: 37px; text-align: center; background-color: #c8c5c4;\">Film</td>\n<td style=\"width: 145px; height: 37px; text-align: center; background-color: #c8c5c4;\">$P(\\text{Film})$</td>\n<td style=\"width: 116px; height: 37px; text-align: center; background-color: #c8c5c4;\">Genre</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">Forgotten Game</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{Avatar}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Sci-Fi</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">The Diamond Valley</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{SW}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Sci-Fi</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">School of Return</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{NYSM}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Thriller</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">The Silk's Nobody</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{TIJ}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Crime</td>\n</tr>\n</tbody>\n</table>\n<p>We are also told that $\\var{no_people}$ people each buy a ticket at the cinema to see a film of their own choosing during this&nbsp;day.</p>\n<h4>a)</h4>\n<p>To calculate the expected number of people who&nbsp;bought tickets&nbsp;to see one of these films we multiply the probability that a person&nbsp;buys a ticket for that film by how many people bought tickets for a film at the cinema.&nbsp;</p>\n<p>So the expected number of people who&nbsp;bought tickets to see <em>Forgotten Game</em> is</p>\n<p>\\[<br/>\\var{Avatar} \\times \\var{no_people} = \\var{{Avatar}*{no_people}}.<br/>\\]</p>\n<h4>b)</h4>\n<p>We are now asked to calculate the expected number of people who bought tickets to see a Sci-Fi film.</p>\n<p>From the table above we can see that there are two films which belong to the Sci-Fi genre: <em>Forgotten Game</em>&nbsp;and <em>The Diamond Valley</em>.</p>\n<p>Firstly, we need to calculate the probability that a person buys a ticket to see a Sci-Fi film, which we will denote $P(\\text{Sci-Fi})$.</p>\n<p>Since the probability that a person buys a ticket to see each film is different, it would be <strong>incorrect</strong> to say that the probability that a person buys a ticket to see a Sci-Fi film is&nbsp;</p>\n<p>\\[\\displaystyle\\frac{2}{4} = \\displaystyle\\frac{1}{2}.\\]</p>\n<p>Instead we must recognise that the probability that a person buys a ticket to see a Sci-Fi film is the probability that a person buys a ticket to see either <em>Forgotten&nbsp;</em><strong>or</strong>&nbsp;<em>The Diamond Valley</em>.</p>\n<p>Therefore to calculate this probability, we add the probabilities of a person buying a ticket to see each of these films:</p>\n<p>\\[<br/>\\begin{align}<br/>P(\\text{Sci-Fi}) &amp;= P(\\text{Forgotten Game})+P(\\text{The Diamond Valley})\\\\<br/>&amp;= \\var{Avatar}+\\var{SW}\\\\<br/>&amp;= \\var{Avatar+SW}.<br/>\\end{align}<br/>\\]</p>\n<p>Then the expected number of people who bought tickets to see a Sci-Fi film is&nbsp;</p>\n<p>\\[<br/>\\var{Avatar+SW} \\times \\var{no_people} = \\var{({Avatar+SW})*{no_people}}.<br/>\\]</p>\n<p></p>", "ungrouped_variables": ["Avatar", "SW", "NYSM", "TIJ", "no_people"], "preamble": {"js": "", "css": ""}, "parts": [{"prompt": "<p>How many of these people would you expect to have bought tickets to see <em>Forgotten Game</em>?</p>", "mustBeReduced": false, "showFeedbackIcon": true, "maxValue": "{no_people}*{Avatar}", "correctAnswerFraction": false, "marks": 1, "type": "numberentry", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "minValue": "{no_people}*{Avatar}", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "variableReplacements": [], "scripts": {}, "allowFractions": false, "correctAnswerStyle": "plain"}, {"prompt": "<p>How many of these people would you expect to have bought tickets to see a Sci-Fi film?</p>", "mustBeReduced": false, "showFeedbackIcon": true, "maxValue": "{no_people}*({Avatar}+{SW})", "correctAnswerFraction": false, "marks": 1, "type": "numberentry", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "minValue": "{no_people}*({Avatar}+{SW})", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "variableReplacements": [], "scripts": {}, "allowFractions": false, "correctAnswerStyle": "plain"}], "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>"}, "variable_groups": [], "extensions": [], "tags": ["Dice", "dice", "Expected values", "Expected Values", "Experimental Probability", "experimental probability", "Experimental probability", "Probability", "probability", "relative frequency", "Relative Frequency", "taxonomy", "Theoretical Probability", "theoretical probability"], "functions": {}, "variables": {"Avatar": {"definition": "random(0.2..0.31 #0.05)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>Probability someone sees Avatar&nbsp;</p>", "name": "Avatar"}, "TIJ": {"definition": "1-(Avatar+SW+NYSM)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>Probability someone goes to see the Italian Job</p>", "name": "TIJ"}, "NYSM": {"definition": "(1-(Avatar+SW))*3/5", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>Probability someone goes to see Now you see me</p>", "name": "NYSM"}, "no_people": {"definition": "random(100..180 #20)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>Number of people who see a movie.</p>", "name": "no_people"}, "SW": {"definition": "random(0.4..0.51 #0.05)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>Probability someone goes to see Star Wars</p>", "name": "SW"}}, "name": "Simon's copy of Calculating Expected Values given a table of probabilities", "statement": "<p>There are&nbsp;four&nbsp;films being shown in a cinema on a particular day.</p>\n<p>The probability that a person buys a ticket to see each&nbsp;film, denoted $P(\\text{Film})$, is given in the table below.</p>\n<table width=\"436\" height=\"248\" border=\"-\" style=\"height: 248px;\">\n<tbody>\n<tr style=\"height: 37px;\">\n<td style=\"width: 126px; height: 37px; text-align: center; background-color: #c8c5c4;\">Film</td>\n<td style=\"width: 145px; height: 37px; text-align: center; background-color: #c8c5c4;\">$P(\\text{Film})$</td>\n<td style=\"width: 116px; height: 37px; text-align: center; background-color: #c8c5c4;\">Genre</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">Forgotten Game</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{Avatar}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Sci-Fi</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">The Diamond Valley</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{SW}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Sci-Fi</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">School of Return</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{NYSM}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Thriller</td>\n</tr>\n<tr style=\"height: 38px;\">\n<td style=\"width: 126px; height: 38px; text-align: center;\">The Silk's Nobody</td>\n<td style=\"width: 145px; height: 38px; text-align: center;\">$\\var{TIJ}$</td>\n<td style=\"width: 116px; height: 38px; text-align: center;\">Crime</td>\n</tr>\n</tbody>\n</table>\n<p>$\\var{no_people}$ people each buy a ticket at the cinema to see a film of their own choosing during the day.</p>", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}