// Numbas version: exam_results_page_options
{"duration": 0, "navigation": {"browse": true, "reverse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true, "showfrontpage": true, "onleave": {"action": "none", "message": ""}}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"allowrevealanswer": true, "showtotalmark": true, "showactualmark": true, "showanswerstate": true, "advicethreshold": 0, "feedbackmessages": [], "intro": ""}, "showstudentname": true, "percentPass": 0, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": " Relative Frequency", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculate relative frequencies in a variety of scenarios.</p>"}, "functions": {}, "ungrouped_variables": ["no_people", "free_range", "no_students", "rel_freq", "dog", "cat", "hamster", "parrot", "n"], "parts": [{"minValue": "{free_range}/{no_people}", "scripts": {}, "customMarkingAlgorithm": "", "precisionMessage": "<p>You must give your answer as a decimal to 2 decimal places.</p>", "mustBeReducedPC": 0, "precisionType": "dp", "showPrecisionHint": true, "prompt": "<p>$\\var{no_people}$ people were asked whether they preferred to buy free-range eggs or caged eggs in supermarkets. $\\var{free_range}$ people said that they preferred to buy free-range eggs. What is the relative frequency of people who prefer buying free-range eggs? Give your answer as a decimal, to $2$ decimal places.&nbsp;</p>\n<p></p>", "type": "numberentry", "variableReplacements": [], "allowFractions": false, "correctAnswerStyle": "plain", "strictPrecision": true, "extendBaseMarkingAlgorithm": true, "marks": 1, "precision": "2", "unitTests": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "maxValue": "{free_range}/{no_people}", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "showCorrectAnswer": true, "precisionPartialCredit": 0}, {"minValue": "{no_students}*{rel_freq}", "scripts": {}, "customMarkingAlgorithm": "", "precisionMessage": "<p>Round your answer to the nearest integer.</p>", "mustBeReducedPC": 0, "precisionType": "dp", "showPrecisionHint": true, "prompt": "<p>The heights of a class of students were measured. The relative frequency of a student being taller than $150$&nbsp;cm&nbsp;is&nbsp;known to be $\\var{rel_freq}$. If there&nbsp;are $\\var{no_students}$ students in the class, estimate the number of students who&nbsp;are taller than $150$ cm.</p>", "type": "numberentry", "variableReplacements": [], "allowFractions": false, "correctAnswerStyle": "plain", "strictPrecision": false, "extendBaseMarkingAlgorithm": true, "marks": 1, "precision": 0, "unitTests": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "maxValue": "{no_students}*{rel_freq}", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "showCorrectAnswer": true, "precisionPartialCredit": 0}, {"marks": 0, "unitTests": [], "variableReplacements": [], "gaps": [{"marks": 1, "minValue": "{dog}+{cat}+{hamster}+{parrot}", "unitTests": [], "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "{dog}+{cat}+{hamster}+{parrot}", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "mustBeReduced": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}, {"minValue": "{dog}/({dog}+{cat}+{hamster}+{parrot})", "scripts": {}, "customMarkingAlgorithm": "", "precisionMessage": "<p>Round your answer to 2 decimal places.</p>", "precisionType": "dp", "showPrecisionHint": true, "precisionPartialCredit": 0, "type": "numberentry", "variableReplacements": [], "allowFractions": false, "correctAnswerStyle": "plain", "strictPrecision": true, "extendBaseMarkingAlgorithm": true, "marks": 1, "precision": "2", "unitTests": [], "correctAnswerFraction": false, "showFeedbackIcon": true, "maxValue": "{dog}/({dog}+{cat}+{hamster}+{parrot})", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "prompt": "<p>A survey was conducted to find out what type of pet is the most common. The results are given in the table below.</p>\n<table style=\"width: 365px; height: 183px;\" border=\"-\" cellspacing=\"1\">\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"width: 104px; height: 17px; text-align: center;\">Type of Pet</td>\n<td style=\"width: 100px; height: 17px; text-align: center;\">Frequency</td>\n</tr>\n<tr style=\"height: 39px;\">\n<td style=\"width: 104px; height: 39px; text-align: center;\">Dog</td>\n<td style=\"width: 100px; height: 39px; text-align: center;\">$\\var{dog}$</td>\n</tr>\n<tr style=\"height: 39px;\">\n<td style=\"width: 104px; height: 39px; text-align: center;\">Cat</td>\n<td style=\"width: 100px; height: 39px; text-align: center;\">$\\var{cat}$</td>\n</tr>\n<tr style=\"height: 43.5px;\">\n<td style=\"width: 104px; height: 43.5px; text-align: center;\">Hamster</td>\n<td style=\"width: 100px; height: 43.5px; text-align: center;\">$\\var{hamster}$</td>\n</tr>\n<tr style=\"height: 17px;\">\n<td style=\"width: 104px; height: 17px; text-align: center;\">Parrot</td>\n<td style=\"width: 100px; height: 17px; text-align: center;\">\n<p style=\"text-align: center;\">$\\var{parrot}$</p>\n</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><em>i)&nbsp;</em></p>\n<p>What was the sample size for the survey?</p>\n<p>[[0]]</p>\n<p><em>ii)</em></p>\n<p>What is the relative frequency of a person having a dog as a pet? Give your answer as a decimal, to $2$ decimal places.</p>\n<p>[[1]]</p>\n<p></p>", "scripts": {}, "type": "gapfill", "showCorrectAnswer": true, "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}], "variable_groups": [], "variables": {"cat": {"definition": "random(10..50 except dog)", "templateType": "anything", "name": "cat", "description": "<p>Frequency of cat in part c.</p>", "group": "Ungrouped variables"}, "free_range": {"definition": "random(10..40)", "templateType": "anything", "name": "free_range", "description": "<p>Number of people who prefer free-range eggs in part a)</p>", "group": "Ungrouped variables"}, "no_students": {"definition": "random(20..40 #10)", "templateType": "anything", "name": "no_students", "description": "<p>Number of students in the class for part b</p>", "group": "Ungrouped variables"}, "hamster": {"definition": "random(10..40 except dog except cat) ", "templateType": "anything", "name": "hamster", "description": "<p>Frequency of hamster in part c</p>", "group": "Ungrouped variables"}, "rel_freq": {"definition": "random(0.1..0.9 # 0.01)", "templateType": "anything", "name": "rel_freq", "description": "<p>Relative frequency for part b</p>", "group": "Ungrouped variables"}, "no_people": {"definition": "random(50..150 #10)", "templateType": "anything", "name": "no_people", "description": "<p>Number of people asked in part a)</p>", "group": "Ungrouped variables"}, "parrot": {"definition": "random(10..50 except dog except cat except hamster)", "templateType": "anything", "name": "parrot", "description": "<p>Frequency of guinea pig in part c.</p>", "group": "Ungrouped variables"}, "n": {"definition": "dog+cat+hamster+parrot", "templateType": "anything", "name": "n", "description": "<p>Sample size for part c</p>", "group": "Ungrouped variables"}, "dog": {"definition": "random(10..50)", "templateType": "anything", "name": "dog", "description": "<p>Frequency of dog in part c</p>", "group": "Ungrouped variables"}}, "rulesets": {}, "preamble": {"js": "", "css": ""}, "tags": [], "statement": "<p>The relative frequency of an outcome is the frequency of the outcome divided by the number of trials.</p>", "advice": "<p>The relative frequency of an outcome is the frequency of the outcome divided by the number of trials.</p>\n<h4>a)</h4>\n<p>We are told that&nbsp;$\\var{no_people}$ people were asked whether they preferred to buy free-range eggs or caged eggs in supermarkets and that&nbsp;$\\var{free_range}$ of these people said that they preferred to buy free-range eggs.&nbsp;</p>\n<p>To calculate the relative frequency of people who prefer buying free-range eggs we need the number of trials and the frequency of people who said that they preferred buying free-range eggs.</p>\n<p>So, the number of trials in this situation is the number of people who were asked the question, which is $\\var{no_people}$.</p>\n<p>The frequency of people who said that they preferred to buy free-range eggs is&nbsp;$\\var{free_range}$.</p>\n<p>Therefore, the relative frequency of people who prefer buying free-range eggs is&nbsp;</p>\n<p>\\[<br/>\\frac{\\var{free_range}}{\\var{no_people}} = \\var{dpformat({free_range/no_people}, 2)} \\; (\\text{rounded to $2$ decimal places}).<br/>\\]&nbsp;</p>\n<h4>b)</h4>\n<p>We are told that the relative frequency of a student being taller than $150$&nbsp;cm is&nbsp;$\\var{rel_freq}$.</p>\n<p>Here, we must use the formula for relative frequency in reverse in order to estimate the number of students in the class who are taller than $150$ cm.</p>\n<p>As we are using relative frequency to calculate this number, our answer may not be completely accurate, therefore our answer will be an estimate of the actual number.</p>\n<p>If we let $n$ denote the number of students in the class who are taller than $150$&nbsp;cm and if there&nbsp;are&nbsp;$\\var{no_students}$ students in the class then</p>\n<p>\\[<br/>\\begin{align}<br/>\\frac{n}{\\var{no_students}} &amp;=&nbsp;\\var{rel_freq}\\\\<br/>n &amp;=&nbsp;\\var{rel_freq} \\times&nbsp;\\var{no_students}\\\\<br/>&amp;= \\var{{rel_freq}*{no_students}}.<br/>\\end{align}<br/>\\]</p>\n<p>As $n$ represents a number of people we must round our value of $n$ to the nearest integer.</p>\n<p>So, the estimated number of students in the class who are taller than $150$&nbsp;cm is $\\var{dpformat({rel_freq}*{no_students},0)}$.</p>\n<h4>c)</h4>\n<p><em> </em></p>\n<p><em>i)&nbsp;</em></p>\n<p>Using the frequency table given in the question, we can calculate the sample size of the survey by adding together the frequencies of each of the different types of pets.</p>\n<p>So, the sample size of the survey is&nbsp;</p>\n<p>\\[<br/>\\var{dog}+\\var{cat}+\\var{hamster}+\\var{parrot} = \\var{{dog}+{cat}+{hamster}+{parrot}}.<br/>\\]</p>\n<p><em>ii)&nbsp;</em></p>\n<p>To calculate the relative frequency of a person having a dog as a pet, we divide the frequency of people in the survey who had a dog as a pet by the sample size of the survey.</p>\n<p>So, the relative frequency is&nbsp;</p>\n<p>\\[<br/>\\frac{\\var{dog}}{\\var{n}} = \\var{dpformat({dog/n}, 2)} \\; (\\text{rounded to $2$ decimal places}).<br/>\\]</p>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}, {"name": "Theoretical Probability vs Experimental Probability ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "type": "question", "statement": "<p>Two unbiased, 6-sided dice were rolled together and the total of the numbers shown on the faces was recorded.</p>\n<p>The experiment was repeated $\\var{no_rolls}$ times.</p>\n<p>This table gives the frequency of each outcome.</p>\n<table border=\"-\" width=\"1344\" height=\"123\">\n<tbody>\n<tr>\n<th style=\"width: 107px; background-color: #c8c5c4;\">Total</th>\n<td style=\"text-align: center; width: 93px;\">2</td>\n<td style=\"text-align: center; width: 93px;\">3</td>\n<td style=\"text-align: center; width: 94px;\">4</td>\n<td style=\"text-align: center; width: 94px;\">5</td>\n<td style=\"text-align: center; width: 94px;\">6</td>\n<td style=\"text-align: center; width: 94px;\">7</td>\n<td style=\"text-align: center; width: 94px;\">8</td>\n<td style=\"text-align: center; width: 94px;\">9</td>\n<td style=\"text-align: center; width: 94px;\">10</td>\n<td style=\"text-align: center; width: 94px;\">11</td>\n<td style=\"text-align: center; width: 106px;\">12</td>\n</tr>\n<tr>\n<th style=\"width: 107px; background-color: #c8c5c4;\">Frequency</th>\n<td style=\"text-align: center; width: 93px;\">$\\var{Freq[0]}$</td>\n<td style=\"text-align: center; width: 93px;\">$\\var{Freq[1]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[2]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[3]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[4]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[5]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[6]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[7]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[8]}$</td>\n<td style=\"text-align: center; width: 94px;\">$\\var{Freq[9]}$</td>\n<td style=\"text-align: center; width: 106px;\">$\\var{Freq[10]}$</td>\n</tr>\n</tbody>\n</table>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"Freq2": {"group": "Ungrouped variables", "name": "Freq2", "description": "<p>List of Frequencies for theoretical probability.</p>", "templateType": "anything", "definition": "[1,2,3,4,5,6,5,4,3,2,1]"}, "sum": {"group": "Ungrouped variables", "name": "sum", "description": "<p>Sums obtained from no_rolls of two dice part a)</p>", "templateType": "anything", "definition": "repeat(random(2..12), no_rolls)"}, "gcd2": {"group": "Ungrouped variables", "name": "gcd2", "description": "<p>Gcd for answer in part a) ii)</p>", "templateType": "anything", "definition": "gcd(Freq2[x[0]],36)"}, "gcd1": {"group": "Ungrouped variables", "name": "gcd1", "description": "<p>Gcd of numerator and denominator for advice for part a) i).</p>", "templateType": "anything", "definition": "gcd(Freq[sum[0]-2], no_rolls)"}, "die": {"group": "Ungrouped variables", "name": "die", "description": "<p>number the die lands on in part a)</p>", "templateType": "anything", "definition": "[2,3,4,5,6,7,8,9,10,11,12]"}, "Freq": {"group": "Ungrouped variables", "name": "Freq", "description": "<p>Frequencies of each possible sum of numbers from rolling 2 die. part a</p>", "templateType": "anything", "definition": "map(\nlen(filter(x=j,x,sum)),\nj, 2..12)"}, "add": {"group": "Ungrouped variables", "name": "add", "description": "", "templateType": "anything", "definition": "random(-5..5 except -1 except 1 except 0)"}, "no_rolls": {"group": "Ungrouped variables", "name": "no_rolls", "description": "<p>Number of rolls of the die in part a.</p>", "templateType": "anything", "definition": "random(50..100 #10)"}, "remainder": {"group": "Ungrouped variables", "name": "remainder", "description": "", "templateType": "anything", "definition": "(ceil(10000/36)+add)+(ceil(10000/18)+add)+(ceil(2500/3)+add)+(ceil(10000/9)+add)+(ceil(12500/9)+add)+(ceil(12500/9)-add)+(ceil(10000/9)-add)+(ceil(2500/3)-add)+(ceil(5000/9)-add)+(ceil(2500/9)-add)"}, "x": {"group": "Ungrouped variables", "name": "x", "description": "<p>Index for part a.</p>", "templateType": "anything", "definition": "indices(die, sum[0])"}}, "functions": {}, "tags": ["dice", "Dice", "Experimental probability", "Experimental Probability", "experimental probability", "Probability", "probability", "sum of two dice", "taxonomy", "theoretical probability", "Theoretical Probability"], "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"correctAnswerFraction": true, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": true, "maxValue": "{Freq[{sum[0]}-2]}/{no_rolls}", "showFeedbackIcon": true, "minValue": "{Freq[{sum[0]}-2]}/{no_rolls}", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showCorrectAnswer": true, "prompt": "<p>Find the experimental probability of rolling a total of $\\var{sum[0]}$.</p>\n<p>Enter your answer as a fraction.</p>\n<p>[[0]]</p>", "marks": 0}, {"correctAnswerFraction": true, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "allowFractions": true, "maxValue": "{Freq2[{x[0]}]}/36", "showFeedbackIcon": true, "prompt": "<p>Now calculate the theoretical probability that the sum of the scores&nbsp;of the two dice is&nbsp;$\\var{sum[0]}$.</p>\n<p>Enter your answer as a fraction.</p>", "minValue": "{Freq2[{x[0]}]}/36", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", ""], "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "choices": ["<p>gets closer to&nbsp;the theoretical probability</p>", "<p>gets further away from&nbsp;the theoretical probability</p>"], "showFeedbackIcon": true, "shuffleChoices": true, "matrix": ["1", 0], "variableReplacements": [], "marks": 0, "displayColumns": 0, "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showCorrectAnswer": true, "prompt": "<p>As the number of rolls of the two dice increases, the experimental probability&nbsp;[[0]]</p>\n<p></p>", "marks": 0}], "ungrouped_variables": ["die", "no_rolls", "sum", "Freq", "Freq2", "x", "gcd1", "gcd2", "add", "remainder"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Compute the experimental probability of a particular score on a die given a sample of throws, and compare it with the theoretical probability.</p>\n<p>The last part asks what you expect to happen to the experimental probability as the sample size increases.</p>"}, "preamble": {"css": "", "js": ""}, "advice": "<p>There are two ways of assigning probability:</p>\n<ul>\n<li><strong>Experimental Probability</strong> is where you do an experiment, observe the outcome and record the relative frequency results.</li>\n<li><strong>Theoretical Probability</strong> is where the probability is calculated based on fairness and symmetrical properties of the results.</li>\n</ul>\n<h4>a)</h4>\n<p><em></em>To calculate the experimental probability (relative frequency) of an outcome we divide the frequency of the outcome in the experiment by the number of trials.</p>\n<p>We are given that the experiment was repeated $\\var{no_rolls}$ times.</p>\n<p>We then need the number of times that the sum of the faces of the dice was equal to $\\var{sum[0]}$. From the frequency table, we can see that the frequency of rolling a $\\var{sum[0]}$ in the experiment was $\\var{Freq[sum[0]-2]}$.</p>\n<p>Therefore, the experimental probability of rolling a $\\var{sum[0]}$ is&nbsp;</p>\n<p><span data-jme-visible=\"gcd(Freq[sum[0]-2], no_rolls) = 1\">\\[ \\begin{align} P(\\text{total}=\\var{sum[0]}) &amp;= \\displaystyle\\frac{\\text{number of times a total of $\\var{sum[0]}$ was rolled}}{\\text{total number of rolls}}\\\\&amp;= \\displaystyle\\frac{\\var{Freq[sum[0]-2]}}{\\var{no_rolls}}.\\end{align}\\]</span></p>\n<p><span data-jme-visible=\"gcd(Freq[sum[0]-2], no_rolls) &gt; 1\">\\[\\begin{align} P(\\text{total}=\\var{sum[0]}) &amp;= \\displaystyle\\frac{\\text{number of times a total of $\\var{sum[0]}$ was rolled}}{\\text{total number of rolls}}\\\\&amp;= \\displaystyle\\frac{\\var{Freq[sum[0]-2]}}{\\var{no_rolls}}\\\\&amp;= \\displaystyle\\var[fractionNumbers, simplifyFractions]{{Freq[sum[0]-2]/no_rolls}}.\\end{align}\\]</span></p>\n<h4>b)</h4>\n<p>When two unbiased 6-sided dice are rolled and their scores are added, there are $11$ possible outcomes: the total must be between $2$ and $12$ inclusive.</p>\n<p>To work out the probabilities of each outcome occurring, we must be very careful about how the experiment is performed.</p>\n<p>There are a total of $36$&nbsp;different outcomes when rolling two dice one after the other; these are shown in Table $1$.</p>\n<table height=\"207\" style=\"width: 375px; height: 235px;\" border=\"-\"><caption>Table 1&nbsp;</caption>\n<tbody>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\"></th>\n<th style=\"width: 38px; text-align: center; background-color: #c8c5c4;\">1</th>\n<th style=\"width: 38px; text-align: center; background-color: #c8c5c4;\">2</th>\n<th style=\"width: 39px; text-align: center; background-color: #c8c5c4;\">3</th>\n<th style=\"width: 39px; text-align: center; background-color: #c8c5c4;\">4</th>\n<th style=\"width: 39px; text-align: center; background-color: #c8c5c4;\">5</th>\n<th style=\"width: 39px; text-align: center; background-color: #c8c5c4;\">6</th>\n</tr>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\">1</th>\n<td style=\"width: 38px; text-align: center;\">(1,1)</td>\n<td style=\"width: 38px; text-align: center;\">(1,2)</td>\n<td style=\"width: 39px; text-align: center;\">(1,3)</td>\n<td style=\"width: 39px; text-align: center;\">(1,4)</td>\n<td style=\"width: 39px; text-align: center;\">(1,5)</td>\n<td style=\"width: 39px; text-align: center;\">(1,6)</td>\n</tr>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\">2</th>\n<td style=\"width: 38px; text-align: center;\">(2,1)</td>\n<td style=\"width: 38px; text-align: center;\">(2,2)</td>\n<td style=\"width: 39px; text-align: center;\">(2,3)</td>\n<td style=\"width: 39px; text-align: center;\">(2,4)</td>\n<td style=\"width: 39px; text-align: center;\">(2,5)</td>\n<td style=\"width: 39px; text-align: center;\">(2,6)</td>\n</tr>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\">3</th>\n<td style=\"width: 38px; text-align: center;\">(3,1)</td>\n<td style=\"width: 38px; text-align: center;\">(3,2)</td>\n<td style=\"width: 39px; text-align: center;\">(3,3)</td>\n<td style=\"width: 39px; text-align: center;\">(3,4)</td>\n<td style=\"width: 39px; text-align: center;\">(3,5)</td>\n<td style=\"width: 39px; text-align: center;\">(3,6)</td>\n</tr>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\">4</th>\n<td style=\"width: 38px; text-align: center;\">(4,1)</td>\n<td style=\"width: 38px; text-align: center;\">(4,2)</td>\n<td style=\"width: 39px; text-align: center;\">(4,3)</td>\n<td style=\"width: 39px; text-align: center;\">(4,4)</td>\n<td style=\"width: 39px; text-align: center;\">(4,5)</td>\n<td style=\"width: 39px; text-align: center;\">(4,6)</td>\n</tr>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\">5</th>\n<td style=\"width: 38px; text-align: center;\">(5,1)</td>\n<td style=\"width: 38px; text-align: center;\">(5,2)</td>\n<td style=\"width: 39px; text-align: center;\">(5,3)</td>\n<td style=\"width: 39px; text-align: center;\">(5,4)</td>\n<td style=\"width: 39px; text-align: center;\">(5,5)</td>\n<td style=\"width: 39px; text-align: center;\">(5,6)</td>\n</tr>\n<tr>\n<th style=\"width: 30px; text-align: center; background-color: #c8c5c4;\">6</th>\n<td style=\"width: 38px; text-align: center;\">(6,1)</td>\n<td style=\"width: 38px; text-align: center;\">(6,2)</td>\n<td style=\"width: 39px; text-align: center;\">(6,3)</td>\n<td style=\"width: 39px; text-align: center;\">(6,4)</td>\n<td style=\"width: 39px; text-align: center;\">(6,5)</td>\n<td style=\"width: 39px; text-align: center;\">(6,6)</td>\n</tr>\n</tbody>\n</table>\n<p>Note that these outcomes are all different.&nbsp;For example,&nbsp;the outcomes (2,1) and (1,2) are not the same because we know the order in which the dice were thrown so we can distinguish between these two outcomes;&nbsp;when the first die lands on $2$ and the second die lands on $1$ the outcome is (2,1), however when the first die lands on $1$ and the second die lands on $2$ the outcome is (1,2).</p>\n<p>The sum of the numbers in these outcomes is the same, we just count them as different outcomes.</p>\n<p>Any one of these $36$ outcomes is equally likely to occur,&nbsp;so the&nbsp;probability of each of these outcomes&nbsp;is $\\displaystyle\\frac{1}{36}$.</p>\n<p>However,&nbsp;if you roll two indistinguishable dice at the same time, you can't differentiate (a 1 and a 2) from (a 2 and a 1). From your point of view, they are the same outcome. The probabilities of the underlying events (each die's score) haven't changed, but the outcomes you observe have.</p>\n<p>In this case you have a total of $21$ outcomes; these outcomes are <em>not</em> all equally likely. There's only one way of obtaining two 1s, while there are two ways of obtaining a 1 and a 2, corresponding to two squares in Table $1$. Hence the probability of obtaining two 1s would be $\\displaystyle\\frac{1}{36}$, whereas the probability of obtaining a 1 and a 2 would be</p>\n<p>\\[\\displaystyle\\frac{2}{36}=\\displaystyle\\frac{1}{18}.\\]</p>\n<p>You&nbsp;<em>can</em> do the experiment this way, but it's easier when you can tell the dice apart.</p>\n<p>Table $2$ shows the corresponding total of the faces of the dice for each of the outcomes in Table $1$.</p>\n<table height=\"207\" style=\"width: 355px; height: 215px;\" border=\"-\"><caption>Table&nbsp;2&nbsp;</caption>\n<tbody>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">+</th>\n<th style=\"width: 35.61px; text-align: center; background-color: #c8c5c4;\">1</th>\n<th style=\"width: 35.61px; text-align: center; background-color: #c8c5c4;\">2</th>\n<th style=\"width: 35.61px; text-align: center; background-color: #c8c5c4;\">3</th>\n<th style=\"width: 35.61px; text-align: center; background-color: #c8c5c4;\">4</th>\n<th style=\"width: 35.61px; text-align: center; background-color: #c8c5c4;\">5</th>\n<th style=\"width: 35.98px; text-align: center; background-color: #c8c5c4;\">6</th>\n</tr>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">1</th>\n<td style=\"width: 35.61px; text-align: center;\">2</td>\n<td style=\"width: 35.61px; text-align: center;\">3</td>\n<td style=\"width: 35.61px; text-align: center;\">4</td>\n<td style=\"width: 35.61px; text-align: center;\">5</td>\n<td style=\"width: 35.61px; text-align: center;\">6</td>\n<td style=\"width: 35.98px; text-align: center;\">7</td>\n</tr>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">2</th>\n<td style=\"width: 35.61px; text-align: center;\">3</td>\n<td style=\"width: 35.61px; text-align: center;\">4</td>\n<td style=\"width: 35.61px; text-align: center;\">5</td>\n<td style=\"width: 35.61px; text-align: center;\">6</td>\n<td style=\"width: 35.61px; text-align: center;\">7</td>\n<td style=\"width: 35.98px; text-align: center;\">8</td>\n</tr>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">3</th>\n<td style=\"width: 35.61px; text-align: center;\">4</td>\n<td style=\"width: 35.61px; text-align: center;\">5</td>\n<td style=\"width: 35.61px; text-align: center;\">6</td>\n<td style=\"width: 35.61px; text-align: center;\">7</td>\n<td style=\"width: 35.61px; text-align: center;\">8</td>\n<td style=\"width: 35.98px; text-align: center;\">9</td>\n</tr>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">4</th>\n<td style=\"width: 35.61px; text-align: center;\">5</td>\n<td style=\"width: 35.61px; text-align: center;\">6</td>\n<td style=\"width: 35.61px; text-align: center;\">7</td>\n<td style=\"width: 35.61px; text-align: center;\">8</td>\n<td style=\"width: 35.61px; text-align: center;\">9</td>\n<td style=\"width: 35.98px; text-align: center;\">10</td>\n</tr>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">5</th>\n<td style=\"width: 35.61px; text-align: center;\">6</td>\n<td style=\"width: 35.61px; text-align: center;\">7</td>\n<td style=\"width: 35.61px; text-align: center;\">8</td>\n<td style=\"width: 35.61px; text-align: center;\">9</td>\n<td style=\"width: 35.61px; text-align: center;\">10</td>\n<td style=\"width: 35.98px; text-align: center;\">11</td>\n</tr>\n<tr>\n<th style=\"width: 27.97px; text-align: center; background-color: #c8c5c4;\">6</th>\n<td style=\"width: 35.61px; text-align: center;\">7</td>\n<td style=\"width: 35.61px; text-align: center;\">8</td>\n<td style=\"width: 35.61px; text-align: center;\">9</td>\n<td style=\"width: 35.61px; text-align: center;\">10</td>\n<td style=\"width: 35.61px; text-align: center;\">11</td>\n<td style=\"width: 35.98px; text-align: center;\">12</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>For equally likely outcomes you can calculate the probability using the formula</p>\n<p>\\[\\text{Probability of an event} = \\displaystyle\\frac{\\text{number of favourable outcomes}}{\\text{total number of outcomes}}.\\]</p>\n<p>Table $3$&nbsp;gives the theoretical probabilities of each of the possible totals of the two dice occurring:</p>\n<table width=\"547\" height=\"67\" style=\"height: 67px;\" border=\"-\"><caption>Table 3&nbsp;</caption>\n<tbody>\n<tr>\n<td style=\"width: 74px; text-align: center; background-color: #c8c5c4;\">Total</td>\n<td style=\"width: 24px; text-align: center;\">2</td>\n<td style=\"width: 25px; text-align: center;\">3</td>\n<td style=\"width: 25px; text-align: center;\">4</td>\n<td style=\"width: 25px; text-align: center;\">5</td>\n<td style=\"width: 25px; text-align: center;\">6</td>\n<td style=\"width: 25px; text-align: center;\">7</td>\n<td style=\"width: 26px; text-align: center;\">8</td>\n<td style=\"width: 26px; text-align: center;\">9</td>\n<td style=\"width: 27px; text-align: center;\">10</td>\n<td style=\"width: 26px; text-align: center;\">11</td>\n<td style=\"width: 26px; text-align: center;\">12</td>\n</tr>\n<tr>\n<td style=\"width: 74px; text-align: center; background-color: #c8c5c4;\">Probability</td>\n<td style=\"width: 24px; text-align: center;\">$\\displaystyle\\frac{1}{36}$</td>\n<td style=\"width: 25px; text-align: center;\">$\\displaystyle\\frac{2}{36}$</td>\n<td style=\"width: 25px; text-align: center;\">$\\displaystyle\\frac{3}{36}$</td>\n<td style=\"width: 25px; text-align: center;\">$\\displaystyle\\frac{4}{36}$</td>\n<td style=\"width: 25px; text-align: center;\">$\\displaystyle\\frac{5}{36}$</td>\n<td style=\"width: 25px; text-align: center;\">$\\displaystyle\\frac{6}{36}$</td>\n<td style=\"width: 26px; text-align: center;\">$\\displaystyle\\frac{5}{36}$</td>\n<td style=\"width: 26px; text-align: center;\">$\\displaystyle\\frac{4}{36}$</td>\n<td style=\"width: 27px; text-align: center;\">$\\displaystyle\\frac{3}{36}$</td>\n<td style=\"width: 26px; text-align: center;\">$\\displaystyle\\frac{2}{36}$</td>\n<td style=\"width: 26px; text-align: center;\">$\\displaystyle\\frac{1}{36}$</td>\n</tr>\n</tbody>\n</table>\n<p><br/><span data-jme-visible=\"gcd(Freq2[x[0]],36) = 1\">\\[P(\\text{total}=\\var{sum[0]}) = \\displaystyle\\frac{\\var{Freq2[x[0]]}}{36}.\\]</span></p>\n<p><span data-jme-visible=\"gcd(Freq2[x[0]],36) &gt; 1\">\\[\\begin{align}&nbsp;P(\\text{total}=\\var{sum[0]}) &amp;= \\displaystyle\\frac{\\var{Freq2[x[0]]}}{36}\\\\ &amp;= \\displaystyle\\var[fractionNumbers,simplifyFractions]{Freq2[x[0]]/36}. \\end{align}\\]</span></p>\n<h4>c)</h4>\n<p>For any experiment, as we make the number of trials very large the experimental probability tends towards the theoretical probability.</p>\n<p>For this experiment, for example, the theoretical probability of the sum of the faces of the two dice being $3$ is</p>\n<p>\\[\\displaystyle\\frac{2}{36} = \\var{sigformat(1/18,3)} \\text{ (rounded to $3$ significant figures)}.\\]</p>\n<p>If we were to roll the two dice&nbsp;a very large number of times, for example $10000$ times, the frequencies of each outcome would be different.</p>\n<p>The table below was produced by the recording the frequency of each outcome after rolling the two dice 10,000 times.</p>\n<table width=\"822\" height=\"68\" style=\"height: 68px;\" border=\"-\">\n<tbody>\n<tr>\n<td style=\"width: 72px; text-align: center; background-color: #c8c5c4;\">Total</td>\n<td style=\"width: 191px; text-align: center;\">2</td>\n<td style=\"width: 191px; text-align: center;\">3</td>\n<td style=\"width: 191px; text-align: center;\">4</td>\n<td style=\"width: 191px; text-align: center;\">5</td>\n<td style=\"width: 191px; text-align: center;\">6</td>\n<td style=\"width: 191px; text-align: center;\">7</td>\n<td style=\"width: 191px; text-align: center;\">8</td>\n<td style=\"width: 191px; text-align: center;\">9</td>\n<td style=\"width: 191px; text-align: center;\">10</td>\n<td style=\"width: 191px; text-align: center;\">11</td>\n<td style=\"width: 200px; text-align: center;\">12</td>\n</tr>\n<tr>\n<td style=\"width: 72px; text-align: center; background-color: #c8c5c4;\">\n<p>Frequency</p>\n</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(10000/36)+{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(10000/18)+{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(2500/3)+{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(10000/9)+{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(12500/9)+{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{10000-{remainder}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(12500/9)-{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(10000/9)-{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(2500/3)-{add}}$</td>\n<td style=\"width: 191px; text-align: center;\">$\\var{ceil(5000/9)-{add}}$</td>\n<td style=\"width: 200px; text-align: center;\">$\\var{ceil(2500/9)-{add}}$</td>\n</tr>\n</tbody>\n</table>\n<p>This means that the new experimental probability of the sum of the faces of the two dice being $3$ is&nbsp;</p>\n<p>\\[\\displaystyle\\frac{\\var{ceil(10000/18) + {add}}}{10000} = &nbsp;\\var{sigformat((ceil(10000/18)+{add})/10000,3)} \\text{ (rounded to $3$ significant figures)}.\\]</p>\n<p>This is very close to the theoretical probability, so we can see how the experimental probability changes as we increases the number of trials.</p>"}, {"name": "Stem and Leaf Plot", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}], "functions": {"join_numbers": {"language": "jme", "parameters": [["numbers", "list"]], "type": "string", "definition": "'^\\\\s*'+join(numbers,'(\\\\s+|\\\\s*,\\\\s*)')+'\\\\s*,?\\\\s*$'"}}, "ungrouped_variables": ["d", "dsort", "d_unit_value"], "variables": {"dsort": {"name": "dsort", "templateType": "anything", "definition": "sort(d)", "group": "Ungrouped variables", "description": ""}, "d_unit_value": {"name": "d_unit_value", "templateType": "anything", "definition": "[dsort[0]-10,dsort[1]-10,dsort[2]-10,dsort[3]-10,dsort[4]-10,dsort[5]-20,dsort[6]-20,dsort[7]-30,dsort[8]-40,dsort[9]-50]", "group": "Ungrouped variables", "description": "<p>Returns list of values of units of numbers in list for the unit part of the stem and leaf diagram.</p>"}, "d": {"name": "d", "templateType": "anything", "definition": "shuffle(repeat(random(10..19),5) + repeat(random(20..29),2)+repeat(random(30..39),1)+repeat(random(40..49),1)+repeat(random(50..59),1))", "group": "Ungrouped variables", "description": ""}}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "{d_unit_value[0]},{d_unit_value[1]},{d_unit_value[2]},{d_unit_value[3]},{d_unit_value[4]}", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "displayAnswer": "{d_unit_value[0]},{d_unit_value[1]},{d_unit_value[2]},{d_unit_value[3]},{d_unit_value[4]}", "type": "patternmatch", "marks": "1", "extendBaseMarkingAlgorithm": true, "matchMode": "regex", "variableReplacements": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "{d_unit_value[5]},{d_unit_value[6]}", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "displayAnswer": "{d_unit_value[5]},{d_unit_value[6]}", "type": "patternmatch", "marks": "1", "extendBaseMarkingAlgorithm": true, "matchMode": "regex", "variableReplacements": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "{d_unit_value[7]}", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "displayAnswer": "{d_unit_value[7]}", "type": "patternmatch", "marks": "1", "extendBaseMarkingAlgorithm": true, "matchMode": "regex", "variableReplacements": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "{d_unit_value[8]}", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "displayAnswer": "{d_unit_value[8]}", "type": "patternmatch", "marks": "1", "extendBaseMarkingAlgorithm": true, "matchMode": "regex", "variableReplacements": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "{d_unit_value[9]}", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "displayAnswer": "{d_unit_value[9]}", "type": "patternmatch", "marks": "1", "extendBaseMarkingAlgorithm": true, "matchMode": "regex", "variableReplacements": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "scripts": {}, "minValue": "10", "mustBeReduced": false, "showFeedbackIcon": true, "unitTests": [], "correctAnswerStyle": "plain", "maxValue": "10", "customMarkingAlgorithm": "", "type": "numberentry", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "scripts": {}, "minValue": "1", "mustBeReduced": false, "showFeedbackIcon": true, "unitTests": [], "correctAnswerStyle": "plain", "maxValue": "1", "customMarkingAlgorithm": "", "type": "numberentry", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "variableReplacements": [], "mustBeReducedPC": 0}], "prompt": "<p>Construct a stem and leaf plot for these data.&nbsp;The \"stem\" is given; you should enter the \"leaves\" in the boxes to the right, separating neighbouring values with a comma.</p>\n<table>\n<tbody>\n<tr>\n<th>1</th>\n<td>[[0]]</td>\n</tr>\n<tr>\n<th>2</th>\n<td>[[1]]</td>\n</tr>\n<tr>\n<th>3</th>\n<td>[[2]]</td>\n</tr>\n<tr>\n<th>4</th>\n<td>[[3]]</td>\n</tr>\n<tr>\n<th>5</th>\n<td>\n<p>[[4]]</p>\n</td>\n</tr>\n</tbody>\n</table>\n<p>What are the stem and leaf units?</p>\n<p>Stem unit =&nbsp;[[5]]</p>\n<p>Leaf unit =&nbsp;[[6]]</p>", "showFeedbackIcon": true, "sortAnswers": false, "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill", "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacements": []}], "metadata": {"licence": "None specified", "description": ""}, "statement": "<p>The following data are&nbsp;points&nbsp;scored by rugby team Newcastle Sparrows in their last $10$ matches.</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{d[0]}$</td>\n<td>$\\var{d[1]}$</td>\n<td>$\\var{d[2]}$</td>\n<td>$\\var{d[3]}$</td>\n<td>$\\var{d[4]}$</td>\n<td>$\\var{d[5]}$</td>\n<td>$\\var{d[6]}$</td>\n<td>$\\var{d[7]}$</td>\n<td>$\\var{d[8]}$</td>\n<td>$\\var{d[9]}$</td>\n</tr>\n</tbody>\n</table>", "tags": [], "advice": "<table>\n<tbody>\n<tr>\n<th><strong>Stem</strong></th>\n<td><strong>Leaves<br/></strong></td>\n<td></td>\n</tr>\n<tr>\n<th>1</th>\n<td>{d_unit_value[0]},{d_unit_value[1]},{d_unit_value[2]},{d_unit_value[3]},{d_unit_value[4]}</td>\n<td style=\"text-align: left;\">Corresponds to {dsort[0]},{dsort[1]},{dsort[2]},{dsort[3]} and {dsort[4]}.</td>\n</tr>\n<tr>\n<th>2</th>\n<td>{d_unit_value[5]},{d_unit_value[6]}</td>\n<td style=\"text-align: left;\">Corresponds to {dsort[5]} and&nbsp;{dsort[6]}</td>\n</tr>\n<tr>\n<th>3</th>\n<td>{d_unit_value[7]}</td>\n<td style=\"text-align: left;\">...and so on.</td>\n</tr>\n<tr>\n<th>4</th>\n<td>{d_unit_value[8]}</td>\n<td></td>\n</tr>\n<tr>\n<th>5</th>\n<td>{d_unit_value[9]}</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p>The stem unit is $10$. The leaf unit is $1$.</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ".question table th {\nborder-right: 1px solid #000;\n}\n.question table td {\n  text-align: left;\n}"}, "rulesets": {}, "type": "question"}]}], "showQuestionGroupNames": false, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Questions on the subtleties and pitfalls of methods of data collection.</p>"}, "name": "Data collection - original without context", "type": "exam", "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}], "extensions": [], "custom_part_types": [], "resources": []}