// Numbas version: finer_feedback_settings
{"name": "Probability - sum of two numbers drawn without replacement", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Probability - sum of two numbers drawn without replacement", "metadata": {"notes": "<p><strong>7/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>Reminded user to input answer as a fraction.</p>\n        <p>Checked calculation.</p>\n        <p>&nbsp;<strong>22/07/2012:</strong></p>\n        <p>Added description.</p>\n        <p>Checked stats extension box.</p>\n        <p><strong>31/07/2012:</strong></p>\n        <p>Question appears to be working correctly.</p>\n        <p><strong>20/12/2012:</strong></p>\n        <p>Checked calculation, OK. Added tested1 tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "\n            \t\t<p>Two numbers are drawn at random without replacement from the numbers m to n.</p>\n            \t\t<p>Find the probability that both are odd given their sum is&nbsp;<em>even</em>.</p>\n            \t\t"}, "showQuestionGroupNames": false, "type": "question", "statement": "\n  <p>Two numbers are drawn at random (and without replacement) from the numbers $\\var{mi}$ to $\\var{ma}$.</p>\n    <p>Find the probability that both numbers are {parity} given that their sum is <em>even</em>.</p>\n  ", "variable_groups": [], "preamble": {"css": "", "js": ""}, "variables": {"numotherpar": {"templateType": "anything", "name": "numotherpar", "group": "Ungrouped variables", "description": "", "definition": "if(parity='even',noodd,noeven)"}, "mi": {"templateType": "anything", "name": "mi", "group": "Ungrouped variables", "description": "", "definition": "random(1..10)"}, "botheven": {"templateType": "anything", "name": "botheven", "group": "Ungrouped variables", "description": "", "definition": "comb(noeven,2)"}, "ma": {"templateType": "anything", "name": "ma", "group": "Ungrouped variables", "description": "", "definition": "mi+random(8..12#2)"}, "parity": {"templateType": "anything", "name": "parity", "group": "Ungrouped variables", "description": "", "definition": "random('odd','even')"}, "mess": {"templateType": "anything", "name": "mess", "group": "Ungrouped variables", "description": "", "definition": "if(gcd(comb(numpar,2),together)=1,'','(after reducing to lowest form as a fraction).')"}, "di": {"templateType": "anything", "name": "di", "group": "Ungrouped variables", "description": "", "definition": "ma-mi"}, "otherparity": {"templateType": "anything", "name": "otherparity", "group": "Ungrouped variables", "description": "", "definition": "if(parity='even','odd','even')"}, "bothodd": {"templateType": "anything", "name": "bothodd", "group": "Ungrouped variables", "description": "", "definition": "comb(noodd,2)"}, "noodd": {"templateType": "anything", "name": "noodd", "group": "Ungrouped variables", "description": "", "definition": "if(isint(mi/2),di/2,di/2+1)"}, "bothsame": {"templateType": "anything", "name": "bothsame", "group": "Ungrouped variables", "description": "", "definition": "if(parity='odd',bothodd,botheven)"}, "noeven": {"templateType": "anything", "name": "noeven", "group": "Ungrouped variables", "description": "", "definition": "if(isint(mi/2),di/2+1,di/2)"}, "numpar": {"templateType": "anything", "name": "numpar", "group": "Ungrouped variables", "description": "", "definition": "if(parity='even',noeven,noodd)"}, "together": {"templateType": "anything", "name": "together", "group": "Ungrouped variables", "description": "", "definition": "botheven+bothodd"}}, "tags": ["MAS1604", "Probability", "checked2015", "conditional probability", "counting", "drawn without replacement", "events", "sampling space", "select without replacement", "sets", "statistics", "subset", "tested1", "urn model", "without replacement"], "parts": [{"prompt": "\n      <p>Probability that both numbers are {parity}= [[0]]</p>\n        <p>Enter your answer as a fraction and not a decimal.</p>\n      ", "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"musthave": {"showStrings": false, "strings": ["/"], "partialCredit": 0, "message": "<p>Input your answer as a fraction</p>"}, "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input your answer as a fraction not a decimal.</p>"}, "type": "jme", "answer": "{bothsame}/{botheven+bothodd}", "answersimplification": "std", "expectedvariablenames": [], "scripts": {}, "showpreview": true, "vsetrange": [0, 1], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrangepoints": 5, "marks": 2}], "scripts": {}, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["parity", "otherparity", "ma", "di", "mess", "mi", "numotherpar", "noodd", "together", "bothodd", "botheven", "numpar", "bothsame", "noeven"], "functions": {}, "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": []}], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "advice": "\n    \n    \n    <p>As we are sampling without replacement the best sampling space is the space of all unordered pairs.</p>\n    \n    \n    \n    <p>This means that when we count up the number of pairs we use the number of ways of selecting pairs.</p>\n    \n    \n    \n    <p>Let  $A$ be the event that both numbers are {parity} and  $B$ the event that their sum is even.</p>\n    \n    \n    \n    <p>Note that $A$ is a subset of  $B$ hence  $P(A \\cap B)=P(A)$.</p>\n    \n    \n    \n    <p>The probability we want to find is $P(A | B)$.</p>\n    \n    \n    \n    <p>Using the definition of conditional probability:</p>\n    \n    \n    \n    <p>\\[P(A | B) = \\frac{P(A \\cap B)}{P(B)} = \\frac{P(A)}{P(B)} \\]</p>\n    \n    \n    \n    <p>Now there are $\\var{numpar}$ {parity} numbers between $\\var{mi}$ and $\\var{ma}$.</p>\n    \n    \n    \n    <p>and as we are sampling without replacement there are</p>\n    \n    \n    \n    <p>\\[{\\var{numpar} \\choose 2} = \\frac{\\var{numpar}\\times \\var{numpar-1}}{2} = \\var{comb(numpar,2)}\\]</p>\n    \n    \n    \n    <p>such pairs, both {parity}.</p>\n    \n    \n    \n    <p>This gives the number of elements in $A$.</p>\n    \n    \n    \n    <p>Also since there are $\\var{ma-mi+1-numpar}$ {otherparity} numbers in the range, there are:</p>\n    \n    \n    \n    <p>\\[{\\var{numotherpar} \\choose 2}=\\var{comb(numotherpar,2)}\\] such pairs, both {otherparity}.</p>\n    \n    \n    \n    <p>There are $\\var{botheven}+\\var{bothodd}=\\var{together}$ pairs with sum even.</p>\n    \n    \n    \n    <p>This gives the number of events in $B$.</p>\n    \n    \n    \n    <p>Hence \\[\\frac{P(A)}{P(B)}=\\frac{\\var{comb(numpar,2)}}{\\var{together}}\\]</p>\n    \n    \n    \n    <p>So the probability that both are {parity} given their sum is even is</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{{comb(numpar,2)}/{together}}\\]</p>\n    \n    \n    \n    <p>{mess}</p>\n    \n    \n  ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}