// Numbas version: exam_results_page_options
{"name": "The probability of an event not happening - five friends play mini golf", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "The probability of an event not happening - five friends play mini golf", "tags": ["complement", "Complement", "complementary", "Probabilities sum to 1", "probability", "Probability"], "metadata": {"description": "<p>Given the probabilities that each of four out of five friends will win a round of mini-golf, work out the probability that the fifth friend won't win, then use that to find the probability that they will win.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Five&nbsp;friends are playing a game of mini-golf.</p>\n<p>The probability that each person wins the game, $\\mathrm{P}(\\text{Person})$,&nbsp;is given in the table.</p>\n<table width=\"816\" height=\"96\" style=\"height: 96px;\" border=\"-\">\n<tbody>\n<tr style=\"height: 27.52px;\">\n<td style=\"width: 119.82px; height: 27.52px; text-align: center; background-color: #c8c5c4;\">Person</td>\n<td style=\"width: 119.82px; height: 27.52px; text-align: center;\">{people[0]['name']}</td>\n<td style=\"width: 119.84px; height: 27.52px; text-align: center;\">{people[1]['name']}</td>\n<td style=\"width: 119.84px; height: 27.52px; text-align: center;\">{people[2]['name']}</td>\n<td style=\"width: 119.84px; height: 27.52px; text-align: center;\">{people[3]['name']}</td>\n<td style=\"width: 119.84px; height: 27.52px; text-align: center;\">{people[4]['name']}</td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 119.82px; height: 17px; text-align: center; background-color: #c8c5c4;\">$\\mathrm{P}(\\text{Person})$</td>\n<td style=\"width: 119.82px; height: 17px; text-align: center;\">$\\var{probs[0]}$</td>\n<td style=\"width: 119.84px; height: 17px; text-align: center;\">$\\var{probs[1]}$</td>\n<td style=\"width: 119.84px; height: 17px;\"></td>\n<td style=\"width: 119.84px; height: 17px; text-align: center;\">$\\var{probs[2]}$</td>\n<td style=\"width: 119.84px; height: 17px; text-align: center;\">$\\var{probs[3]}$</td>\n</tr>\n</tbody>\n</table>", "advice": "<p>All probability situations can be reduced to two possible outcomes: success or failure.</p>\n<p>When we express the outcomes in this way we say that they are complementary.</p>\n<p>The sum of the probability of an event and its complement is always $1$.</p>\n<p>If $\\mathrm{P}(\\mathrm{E})$ is the probability of an event $\\mathrm{E}$ happening and $\\mathrm{P}(\\bar{\\mathrm{E}})$ is the probability of that event not happening then</p>\n<p>\\[\\mathrm{P}(\\mathrm{E}) +\\mathrm{P}(\\bar{\\mathrm{E}}) = 1.\\]</p>\n<p>Rearranging this equation gives:</p>\n<p>\\[\\mathrm{P}(\\bar{\\mathrm{E}}) = 1 - \\mathrm{P}(\\mathrm{E})\\]</p>\n<p>We can think of this game as having two possible outcomes: either {pname} wins or {pname} doesn't win.</p>\n<p>This means that</p>\n<p>\\[\\mathrm{P}(\\var{pname}) + \\mathrm{P}(\\text{not } \\var{pname}) = 1 \\text{.}\\]</p>\n<h4></h4>\n<h4>a)</h4>\n<p><em></em>If {pname} doesn't win the game then that means that one of the other four players must win the game.</p>\n<p>So the probability of {pname} not winning the game is the same as the probability of any of the other four players winning the game.</p>\n<p>Therefore</p>\n<p>\\begin{align}<br/>\\mathrm{P}(\\text{not }\\var{pname}) &amp;= \\mathrm{P}(\\var{people[0]['name']})+\\mathrm{P}(\\var{people[1]['name']})+\\mathrm{P}(\\var{people[3]['name']})+\\mathrm{P}(\\var{people[4]['name']}) \\\\<br/>&amp;= \\var{latex(join(probs,' + '))}\\\\<br/>&amp;= \\var{sum(probs)}.<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>Rearranging the equation above gives</p>\n<p>\\[\\mathrm{P}(\\var{pname}) = 1 - \\mathrm{P}(\\text{not } \\var{pname}).\\]</p>\n<p>We know from <strong>a)</strong> that $\\mathrm{P}(\\text{not } \\var{pname}) = \\var{sum(probs)}$.</p>\n<p>Therefore</p>\n<p>\\begin{align}<br/>\\mathrm{P}(\\var{pname}) &amp;= 1 - \\mathrm{P}(\\text{not } \\var{pname})\\\\<br/>&amp;= 1 - \\var{sum(probs)}\\\\<br/>&amp;= \\var{1-sum(probs)}.<br/>\\end{align}</p>", "rulesets": {}, "extensions": ["random_person"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"probs": {"name": "probs", "group": "Ungrouped variables", "definition": "map(precround(raw_probs[j]/sum(raw_probs),2),j,0..3)", "description": "<p>The probability of each of the first 4 friends winning the game. The missing person isn't included, so their probability can be 1 minus the sum of the rest, accumulating any rounding errors.</p>", "templateType": "anything", "can_override": false}, "pname": {"name": "pname", "group": "Ungrouped variables", "definition": "person['name']", "description": "", "templateType": "anything", "can_override": false}, "person": {"name": "person", "group": "Ungrouped variables", "definition": "people[2]", "description": "<p>The person whose probability is not given.</p>", "templateType": "anything", "can_override": false}, "raw_probs": {"name": "raw_probs", "group": "Ungrouped variables", "definition": "repeat(random(0..1#0),5)", "description": "<p>Uniform random values for each of the five friends. Their winning probabilities will be in proportion to this.</p>", "templateType": "anything", "can_override": false}, "people": {"name": "people", "group": "Ungrouped variables", "definition": "random_people(5)", "description": "", "templateType": "anything", "can_override": false}, "p_not_name": {"name": "p_not_name", "group": "Ungrouped variables", "definition": "sum(probs)", "description": "<p>The probability that the chosen person does not win.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["people", "raw_probs", "probs", "person", "pname", "p_not_name"], "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>What is $\\mathrm{P}(\\text{not } \\var{pname})$?</p>\n<p>[[0]]</p>\n<p><em></em></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "P(not {name})", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p_not_name", "maxValue": "p_not_name", "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><em></em>What is $\\mathrm{P}(\\var{pname})$?</p>\n<p>[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "P({name})", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "p_not_name", "part": "p0g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "1-p_not_name", "maxValue": "1-p_not_name", "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": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}]}