// Numbas version: exam_results_page_options
{"name": "Find expected profit of gambles, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"profit": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(numberbets*bet-numberbets*bet*(odds1+odds2)/(odds2*number),2)", "description": "", "name": "profit"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..6)", "description": "", "name": "r"}, "number": {"templateType": "anything", "group": "Ungrouped variables", "definition": "37", "description": "", "name": "number"}, "odds2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "odds2"}, "bet": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,10,50,100)", "description": "", "name": "bet"}, "odds1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "35", "description": "", "name": "odds1"}, "numberbets": {"templateType": "anything", "group": "Ungrouped variables", "definition": "10^r", "description": "", "name": "numberbets"}}, "ungrouped_variables": ["profit", "numberbets", "number", "r", "bet", "odds2", "odds1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find expected profit of gambles, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "profit", "minValue": "profit", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n      <p>Expected profit=&pound;[[0]]</p>\n        <p>Enter to two decimal places.</p>\n        <p></p>\n      ", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>A roulette table has $\\var{number}$ numbers and pays at $\\var{odds1}$ to $\\var{odds2}$ if the winning number is chosen.</p>\n  <p>Find the expected profit to the casino if $\\var{10^{r}}$ bets of &pound;$\\var{bet}$ are placed independently.</p>", "tags": ["checked2015", "MAS8380", "MAS8401"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>04/11/2013</strong></p>\n    <p>Fix typo \"fod\" -&gt; \"find\".</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a large number of gambles, find the expected profit.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n  <p>The probability of winning is $\\dfrac{1}{\\var{number}}$ and the odds of $\\var{odds1}$ to $\\var{odds2}$ tells us that each winning choice realises \\[\\text{&pound;}\\simplify{{odds1+odds2}/{odds2}}\\times \\var{bet}=\\text{&pound;}\\var{(odds1+odds2)*bet/odds2}\\]on a bet of &pound; $\\var{bet}$.&nbsp;</p>\n    <p>Hence the expected payout on a bet of &pound;$\\var{bet}$ is &pound;$\\frac{\\var{(odds1+odds2)*bet}}{\\var{odds2*number}}$</p>\n    <p>So the expected payout on $\\var{numberbets}$ bets of &pound;$\\var{bet}$ is $\\var{numberbets}\\times \\frac{\\var{(odds1+odds2)*bet}}{\\var{odds2*number}}$</p>\n    <p>Hence:</p>\n    <p>Profit = Income - Payout</p>\n    <p>$=\\text{&pound;}\\var{numberbets}\\times \\var{bet}-\\var{numberbets}\\times \\frac{\\var{(odds1+odds2)*bet}}{\\var{odds2*number}}= \\text{&pound;}\\var{profit}$ to 2 decimal places.</p>\n    <p></p>\n  ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}