// Numbas version: finer_feedback_settings
{"name": "PC2 - Entropy", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "PC2 - Entropy", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>We have information about nests and the number of eggs that they contain. The information is summarised in the following table.</p>\n<table>\n<tbody>\n<tr>\n<td>Clutch Size</td>\n<td>$3$</td>\n<td>$4$</td>\n<td>$5$</td>\n<td>$6$</td>\n<td>$7$</td>\n<td>$8$</td>\n<td>$9$</td>\n</tr>\n<tr>\n<td>$P(C=c)$</td>\n<td>{p3}</td>\n<td>{p4}</td>\n<td>{p5}</td>\n<td>{p6}</td>\n<td>{p7}</td>\n<td>{p8}</td>\n<td>{p9}</td>\n</tr>\n</tbody>\n</table>", "advice": "<p><strong>a)</strong> The formula for Shannon entropy is:</p>\n<p>\\begin{align}<br/>H(x) &amp;= -\\sum_{i=1}^n P(x_i)\\cdot ln(P(x_i)).<br/>\\end{align}</p>\n<p>Calculating this using the information about Clutch sizes in the table gives,</p>\n<p>\\begin{align}<br/>H(x) &amp;= -\\left(\\var{p3}\\cdot ln(\\var{p3}) + \\var{p4}\\cdot ln(\\var{p4}) + \\var{p5}\\cdot ln(\\var{p5}) + \\var{p6}\\cdot ln(\\var{p6}) + \\var{p7}\\cdot ln(\\var{p7}) + \\var{p8}\\cdot ln(\\var{p8}) + \\var{p9}\\cdot ln(\\var{p9})\\right) \\\\<br/>&amp;= \\var{shanent}.<br/>\\end{align}</p>\n<p>Hence, $H(x) = \\var{round_shanent}$ to two decimal places.</p>\n<p><strong>b)&nbsp;</strong>An even clutch size will mean the nest contains $4$, $6$, or $8$ eggs. Therefore,&nbsp;</p>\n<p>\\begin{align}<br/>P(c\\text{ is even}) &amp;= P(C=4) + P(C=6) + P(c=8) \\\\<br/>&amp;= \\var{p4} + \\var{p6} + \\var{p8} \\\\<br/>&amp;= \\var{even}.<br/>\\end{align}</p>\n<p><strong>c)</strong> If we know the clutch size is even we immediately know that the probability that the nests contain an odd number of eggs will be 0. Hence,&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>Clutch Size</td>\n<td>$3$</td>\n<td>$4$</td>\n<td>$5$</td>\n<td>$6$</td>\n<td>$7$</td>\n<td>$8$</td>\n<td>$9$</td>\n</tr>\n<tr>\n<td>$P(C=c|C\\text{ is even})$</td>\n<td>$0$</td>\n<td></td>\n<td>$0$</td>\n<td></td>\n<td>$0$</td>\n<td></td>\n<td>$0$</td>\n</tr>\n</tbody>\n</table>\n<p>Now, using the formula for conditional probability,&nbsp;</p>\n<p>\\begin{align}<br/>P(C=c|C\\text{ is even}) &amp;= \\frac{P(C=c\\text{ and }C\\text{ is even})}{P(C\\text{ is even})}.<br/>\\end{align}</p>\n<p>So, for $c=4$,</p>\n<p>\\begin{align}<br/>P(C=4|C\\text{ is even}) &amp;= \\frac{P(C=4)}{P(C\\text{ is even})} \\\\<br/>&amp;= \\frac{\\var{p4}}{\\var{even}} \\\\<br/>&amp;= \\var{p4e2}, \\text{ to two decimal places}.<br/>\\end{align}</p>\n<p>We use the same formula for $c=6$ and $c=8$,&nbsp;</p>\n<p>\\begin{align}<br/>P(C=6|C\\text{ is even}) &amp;= \\frac{P(C=6)}{P(C\\text{ is even})} \\\\<br/>&amp;= \\frac{\\var{p6}}{\\var{even}} \\\\<br/>&amp;= \\var{p6e2}, \\text{ to two decimal places} \\\\<br/>&amp; \\text{and} \\\\<br/>P(C=8|C\\text{ is even}) &amp;= \\frac{P(C=8)}{P(C\\text{ is even})} \\\\<br/>&amp;= \\frac{\\var{p8}}{\\var{even}} \\\\<br/>&amp;= \\var{p8e2}, \\text{ to two decimal places}.<br/>\\end{align}</p>\n<p>So our completed probability distribution is:</p>\n<table>\n<tbody>\n<tr>\n<td>Clutch Size</td>\n<td>$3$</td>\n<td>$4$</td>\n<td>$5$</td>\n<td>$6$</td>\n<td>$7$</td>\n<td>$8$</td>\n<td>$9$</td>\n</tr>\n<tr>\n<td>$P(C=c|C\\text{ is even})$</td>\n<td>$0$</td>\n<td>{p4e2}</td>\n<td>$0$</td>\n<td>{p6e2}</td>\n<td>$0$</td>\n<td>{p8e2}</td>\n<td>$0$</td>\n</tr>\n</tbody>\n</table>\n<p><strong>d)</strong> We calculate the Shannon entropy utilising the same formula from part&nbsp;<strong>b</strong>,</p>\n<p>\\begin{align}<br/>H(x) &amp;= -\\left(\\var{p4e2}\\cdot ln(\\var{p4e2}) + \\var{p6e2}\\cdot ln(\\var{p6e2}) + \\var{p8e2}\\cdot ln(\\var{p8e2}) \\right) \\\\<br/>&amp;= \\var{even_shanent}.<br/>\\end{align}</p>\n<p>Hence, $H(x) = \\var{roundeven_shanent}$, rounded to two decimal places.</p>\n<p><strong>e)</strong> Since the entropy decreases when we learn that the clutch size is even, we can say that we are more certan about the outcome.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"p3": {"name": "p3", "group": "Ungrouped variables", "definition": "if(randnum=1,0.01,if(randnum=2,0.02,if(randnum=3,0.02,0)))", "description": "<p>probability of the nest having 3 eggs</p>", "templateType": "anything", "can_override": false}, "p4": {"name": "p4", "group": "Ungrouped variables", "definition": "if(randnum=1,0.17,if(randnum=2,0.13,if(randnum=3,0.17,0)))", "description": "<p>probability of the nest having 4 eggs</p>", "templateType": "anything", "can_override": false}, "p5": {"name": "p5", "group": "Ungrouped variables", "definition": "if(randnum=1,0.35,if(randnum=2,0.4,if(randnum=3,0.34,0)))", "description": "<p>probability of the nest having 5 eggs</p>", "templateType": "anything", "can_override": false}, "p6": {"name": "p6", "group": "Ungrouped variables", "definition": "if(randnum=1,0.33,if(randnum=2,0.32,if(randnum=3,0.3,0)))", "description": "<p>probability of the nest having 6 eggs</p>", "templateType": "anything", "can_override": false}, "p7": {"name": "p7", "group": "Ungrouped variables", "definition": "if(randnum=1,0.09,if(randnum=2,0.1,if(randnum=3,0.08,0)))", "description": "<p>probability of the nest having 7 eggs</p>", "templateType": "anything", "can_override": false}, "p8": {"name": "p8", "group": "Ungrouped variables", "definition": "if(randnum=1,0.03,if(randnum=2,0.02,if(randnum=3,0.06,0)))", "description": "<p>probability of the nest having 8 eggs</p>", "templateType": "anything", "can_override": false}, "p9": {"name": "p9", "group": "Ungrouped variables", "definition": "if(randnum=1,0.02,if(randnum=2,0.01,if(randnum=3,0.03,0)))", "description": "<p>probability of the nest having 9 eggs</p>", "templateType": "anything", "can_override": false}, "randnum": {"name": "randnum", "group": "Ungrouped variables", "definition": "random(1,2,3,1,2,3)", "description": "<p>Random number generator to decide which set of probabilities the question will use</p>", "templateType": "anything", "can_override": false}, "shanent": {"name": "shanent", "group": "Ungrouped variables", "definition": "-(p3*ln(p3)+p4*ln(p4)+p5*ln(p5)+p6*ln(p6)+p7*ln(p7)+p8*ln(p8)+p9*ln(p9))", "description": "<p>Calculation of shannon entropy</p>", "templateType": "anything", "can_override": false}, "round_shanent": {"name": "round_shanent", "group": "Ungrouped variables", "definition": "precround(shanent,2)", "description": "<p>Shannon entropy rounded to 2 decimal places</p>", "templateType": "anything", "can_override": false}, "even": {"name": "even", "group": "Even Clutch Size (Part b & c)", "definition": "p4+p6+p8", "description": "<p>The probability that the clutch size is even</p>", "templateType": "anything", "can_override": false}, "p4e": {"name": "p4e", "group": "Even Clutch Size (Part b & c)", "definition": "p4/even", "description": "<p>probability clutch size has 4 eggs given clutch size is even</p>", "templateType": "anything", "can_override": false}, "p6e": {"name": "p6e", "group": "Even Clutch Size (Part b & c)", "definition": "p6/even", "description": "<p>probability clutch size has 6 eggs given clutch size is even</p>", "templateType": "anything", "can_override": false}, "p8e": {"name": "p8e", "group": "Even Clutch Size (Part b & c)", "definition": "p8/even", "description": "<p>probability clutch size has 8 eggs given clutch size is even</p>", "templateType": "anything", "can_override": false}, "p4e2": {"name": "p4e2", "group": "Even Clutch Size (Part b & c)", "definition": "precround(p4e,2)", "description": "<p>probability clutch size has 4 eggs given clutch size is even, rounded to 2 d.p.</p>", "templateType": "anything", "can_override": false}, "p6e2": {"name": "p6e2", "group": "Even Clutch Size (Part b & c)", "definition": "precround(p6e,2)", "description": "<p>probability clutch size has 6 eggs given clutch size is even, rounded to 2 d.p.</p>", "templateType": "anything", "can_override": false}, "p8e2": {"name": "p8e2", "group": "Even Clutch Size (Part b & c)", "definition": "precround(p8e,2)", "description": "<p>probability clutch size has 8 eggs given clutch size is even, rounded to 2 d.p.</p>", "templateType": "anything", "can_override": false}, "even_shanent": {"name": "even_shanent", "group": "Even Clutch Size (Part b & c)", "definition": "-(p4e*ln(p4e)+p6e*ln(p6e)+p8e*ln(p8e))", "description": "<p>Shannon entropy</p>", "templateType": "anything", "can_override": false}, "roundeven_shanent": {"name": "roundeven_shanent", "group": "Even Clutch Size (Part b & c)", "definition": "precround(even_shanent,2)", "description": "<p>Shannon entropy, rounded to 2 decimla places</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p3", "p4", "p5", "p6", "p7", "p8", "p9", "randnum", "shanent", "round_shanent"], "variable_groups": [{"name": "Even Clutch Size (Part b & c)", "variables": ["even", "p4e", "p4e2", "p6e", "p6e2", "p8e", "p8e2", "even_shanent", "roundeven_shanent"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": true, "customName": "a)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the Shannon entropy (in nats) of the clutch size.</p>\n<p><em>Round your answer to two decimal places if necessary.</em></p>", "minValue": "round_shanent", "maxValue": "round_shanent", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "b)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the probability that the clutch size is even i.e., $P(c\\text{ is even})$?</p>", "minValue": "even", "maxValue": "even", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "gapfill", "useCustomName": true, "customName": "c)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If we know the clutch size is an even number, compute the new distribution $P(C=c|c\\text{ is even})$.</p>\n<p><em>Round your answers to two decimal places where necessary.</em></p>\n<table style=\"height: 42px;\">\n<tbody>\n<tr style=\"height: 20px;\">\n<td style=\"width: 204px; height: 20px;\">Clutch Size</td>\n<td style=\"width: 35px; height: 20px;\">$3$</td>\n<td style=\"width: 25px; height: 20px;\">$4$</td>\n<td style=\"width: 25px; height: 20px;\">$5$</td>\n<td style=\"width: 25px; height: 20px;\">$6$</td>\n<td style=\"width: 25px; height: 20px;\">$7$</td>\n<td style=\"width: 25px; height: 20px;\">$8$</td>\n<td style=\"width: 25px; height: 20px;\">$9$</td>\n</tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 204px; height: 22px;\">$P(C=c|C\\text{ is even})$</td>\n<td style=\"width: 35px; height: 22px;\">[[0]]</td>\n<td style=\"width: 25px; height: 22px;\">[[1]]</td>\n<td style=\"width: 25px; height: 22px;\">[[2]]</td>\n<td style=\"width: 25px; height: 22px;\">[[3]]</td>\n<td style=\"width: 25px; height: 22px;\">[[4]]</td>\n<td style=\"width: 25px; height: 22px;\">[[5]]</td>\n<td style=\"width: 25px; height: 22px;\">[[6]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p4e2", "maxValue": "p4e2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "5", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "6", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p6e2", "maxValue": "p6e2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "7", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "8", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p8e2", "maxValue": "p8e2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "9", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "numberentry", "useCustomName": true, "customName": "d)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Recalculate the Shannon Entropy using the distribution you found in part&nbsp;<strong>c</strong>.</p>\n<p><em>Round your answer to 2 decimal places.</em></p>", "minValue": "roundeven_shanent", "maxValue": "roundeven_shanent", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": true, "customName": "e)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Comparing your Shannon entropy with the one you calculated in part&nbsp;<strong>b</strong>, which statement in true about our sysem?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["We are&nbsp;<strong>more</strong> certain about the outcome i.e., the number of eggs in a nest.", "We are&nbsp;<strong>less</strong> certain about the outcome i.e., the number of eggs in a nest."], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "resources": []}]}], "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}]}