// Numbas version: exam_results_page_options
{"name": "Probability of scoring at basketball given probability of not scoring", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"js": "", "css": ""}, "type": "question", "name": "Probability of scoring at basketball given probability of not scoring", "parts": [{"correctAnswerFraction": true, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "allowFractions": true, "scripts": {}, "minValue": "1-1/{Hoop}", "maxValue": "1-1/{Hoop}", "marks": 1, "variableReplacements": [], "prompt": "<p>What is the probability that the ball goes through the hoop?</p>\n<p><em>Input your answer as a fraction.</em></p>"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given the probability that a basketball shot misses the hoop, find the probability that it's on target - use the law of total probability.</p>"}, "tags": ["taxonomy"], "variables": {"Hoop": {"templateType": "anything", "description": "<p>Denominator of the fraction for the probability that the ball misses the hoop.</p>", "definition": "random(5..9)", "name": "Hoop", "group": "Ungrouped variables"}}, "rulesets": {}, "extensions": [], "functions": {}, "ungrouped_variables": ["Hoop"], "statement": "<p>You are playing a game of&nbsp;basketball against your friend. You have the ball but your friend is blocking you from moving forwards so you throw the ball and hope that it goes through the hoop.</p>\n<p>The probability that the ball misses the hoop is $\\displaystyle \\frac{1}{\\var{Hoop}}$.</p>", "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>When we throw the ball&nbsp;we can say that there&nbsp;are two possible outcomes: either the ball goes through the hoop or the ball does not go through the hoop (the ball misses the hoop).</p>\n<p>Let $\\mathrm{H}$ be the event that the ball goes through the hoop. Then</p>\n<p>\\[\\mathrm{P}(\\mathrm{H}) + \\mathrm{P}(\\bar{\\mathrm{H}}) = 1.\\]</p>\n<p>But we are given that $\\mathrm{P}(\\bar{\\mathrm{H}}) = \\displaystyle\\frac{1}{\\var{Hoop}}$.</p>\n<p>Rearranging the above equation&nbsp;to obtain $\\mathrm{P}(\\mathrm{H})$.</p>\n<p>\\begin{align}<br/>\\mathrm{P}(\\mathrm{H}) &amp;= 1 - \\mathrm{P}(\\bar{\\mathrm{H}}) \\\\[0.5em]<br/>&amp;= 1 - \\displaystyle\\frac{1}{\\var{Hoop}}\\\\[0.5em]<br/>&amp;= \\simplify[fractionNumbers]{{1-1/{Hoop}}}.<br/>\\end{align}</p>\n<h4></h4>", "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}