// Numbas version: exam_results_page_options
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P(A)$.</p>\n<p>Therefore $P(A') = 1 - \\var{A} = \\var{1-A}$.</p>\n<p></p>\n<p>b) $P(A \\cup B)$ is the probability of \"$A$ union $B$\" i.e. the probability of either $A$, or $B$, or both.<br/><br/>You can use the Venn diagram to verify the formula $P(A \\cup B) = P(A) + P(B) - P(A \\cap B)$.<br/><br/>Therefore&nbsp;$P(A \\cup B) = P(A) + P(B) - P(A \\cap B) = \\var{A} + \\var{B} - \\var{both} = \\var{A+B-both}$.</p>\n<p></p>\n<p>c) $P(A \\cap B')$ is the probability of \"$A$ intersection not $B$\" i.e. the probability of both $A$ and $B'$, or 'just $A$'.</p>\n<p>You can use the Venn diagram to confirm that $P(A \\cap B') = P(A) - P(A \\cap B) = \\var{A} - \\var{both} = \\var{A-both}$.</p>\n<p><br/><br/></p>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "random(0.01 .. 0.99#0.01)", "description": "", "templateType": "randrange", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(0.01 .. 0.99#0.01)", "description": "", "templateType": "randrange", "can_override": false}, "justA": {"name": "justA", "group": "Ungrouped variables", "definition": "A-both", "description": "", "templateType": "anything", "can_override": false}, "justB": {"name": "justB", "group": "Ungrouped variables", "definition": "B-both", "description": "", "templateType": "anything", "can_override": false}, "both": {"name": "both", "group": "Ungrouped variables", "definition": "A+B-AuB", "description": "", "templateType": "anything", "can_override": false}, "AuB": {"name": "AuB", "group": "Ungrouped variables", "definition": "1-neither", "description": "", "templateType": "anything", "can_override": false}, "neither": {"name": "neither", "group": "Ungrouped variables", "definition": "1-random(max(A,B)..1 #0.01)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "both>0", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "justA", "justB", "both", "AuB", "neither"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $P(A')$</p>", "minValue": "1-{A}", "maxValue": "1-{A}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $P(A\\cup B)$</p>", "minValue": "{AuB}", "maxValue": "{AuB}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $P(A \\cap B')$</p>", "minValue": "{justA}", "maxValue": "{justA}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}