// Numbas version: finer_feedback_settings
{"name": "Probability: Independent Events - requires 'given that'", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Probability: Independent Events - requires 'given that'", "tags": [], "metadata": {"description": "<p>Independent events in probability. Choose whether given three given pairs of events are independent or not.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Choose whether the following three pairs of events are independent or not.</p>", "advice": "<p>There are two ways to understand independent events, either by thinking about the context of the events, or by examining the mathematical properties.</p>\n<p>If two events are independent we understand that <em>information about one event does not affect our understanding of the likelihood of the other event.<br/></em>e.g. 'it is raining', 'I carry an umberella' are NOT independent, they are dependent.</p>\n<p>To translate this thought into a mathematical statement:</p>\n<p>If A and B are independent events then $P(A \\vert B) = P(A)$.<br/>Since we know (by definition) that $P(A \\vert B) = \\frac{P(A \\cap B)}{P(B)}$, we have that, for independent events,</p>\n<p>$P(A) =&nbsp;P(A \\vert B) = \\frac{P(A \\cap B)}{P(B)}$</p>\n<p>$\\implies &nbsp;P(A) =&nbsp;\\frac{P(A \\cap B)}{P(B)}$</p>\n<p>$\\implies&nbsp;P(A \\cap B) = P(A) \\times P(B)&nbsp;$</p>\n<p>This equality only holds for independent events, and can be used to check if events are independent or not.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0.1..0.9 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(0.1..0.9 #0.1 except a)", "description": "", "templateType": "anything", "can_override": false}, "anb": {"name": "anb", "group": "Ungrouped variables", "definition": "if(ind[0]=1,a*b,random(0.1..max(a,b) #0.1 except a*b))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": 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"ee", "group": "Ungrouped variables", "definition": "random(0.1..0.8 #0.1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ind", "a", "b", "anb", "c", "d", "cnd", "ee", "f", "egivf"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "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>Given that:</p>\n<p>$P(A) = \\var{a}, P(B) = \\var{b}$ and $P(A \\cap B) = \\var{anb}$</p>\n<p></p>\n<p>$P(C) = \\var{c}, P(D) = \\var{d}$ and $P(C \\cap D) = \\var{cnd}$</p>\n<p></p>\n<p>$P(E) = \\var{ee}, P(F) = \\var{f}$ and $P(E&nbsp;\\vert&nbsp;F) = \\var{egivf}$<br/><br/><br/>Which pairs of events are independent?</p>", "minMarks": 0, "maxMarks": "3", "minAnswers": "3", "maxAnswers": "3", "shuffleChoices": false, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$A$ and $B$", "$C$ and $D$", "$E$ and $F$"], "matrix": [["if(ind[0]=0,0,1)", "if(ind[0]=1,0,1)"], ["if(ind[1]=0,0,1)", "if(ind[1]=1,0,1)"], ["if(ind[2]=0,0,1)", "if(ind[2]=1,0,1)"]], "layout": {"type": "all", "expression": ""}, "answers": ["Independent", "Dependent"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ruth Hand", "profile_url": 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