// Numbas version: exam_results_page_options
{"name": "Probability mass function of string of heads from weighted coin, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Probability mass function of string of heads from weighted coin, ", "tags": ["checked2015", "counting", "discrete distribution", "discrete random variable", "expectation", "Probability", "probability", "probability mass function", "random variab;e", "runs", "sample space", "statistics", "variance"], "metadata": {"description": "<p>A weighted coin with given $P(H),\\;P(T)$ is tossed 3 times. Let $X$ be the random variable which denotes the longest string of consecutive heads that occur during these tosses. Find the Probability Mass Function (PMF), expectation and variance of $X$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A coin has two faces, $H$ or $T$. It is weighted so that $P(H)=\\var{h}$ and $P(T)=\\var{t}$ and is tossed $\\var{thismany}$ times.&nbsp;</p>\n<p>Let $X$ be the random variable which denotes the longest string of consecutive heads that occur during these tosses.</p>\n<p>Compute the:</p>\n<ul>\n<li>Probability Mass Function (PMF) of $X$.</li>\n<li>Expectation of $X$.</li>\n<li>Variance of $X$.</li>\n</ul>", "advice": "<p>a)</p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td align=\"left\"><strong>Event</strong></td>\n<th><strong>HHH</strong></th>\n<th><strong>HHT</strong></th>\n<th><strong>HTH</strong></th>\n<th><strong>HTT</strong></th>\n</tr>\n<tr>\n<td align=\"left\"><strong>Probability</strong></td>\n<td>$\\var{h}^3=\\var{h^3}$</td>\n<td>$\\var{h}^2\\times\\var{t}=\\var{h^2*t}$</td>\n<td>$\\var{h}^2\\times\\var{t}=\\var{h^2*t}$</td>\n<td>$\\var{h}\\times\\var{t}^2=\\var{h*t^2}$</td>\n</tr>\n<tr>\n<td align=\"left\"><strong>$X$</strong></td>\n<td>$3$</td>\n<td>$2$</td>\n<td>$1$</td>\n<td>$1$</td>\n</tr>\n</tbody>\n</table>\n<table border=\"2\">\n<tbody>\n<tr>\n<td align=\"left\"><strong>Event</strong></td>\n<th><strong>THH</strong></th>\n<th><strong>THT</strong></th>\n<th><strong>TTH</strong></th>\n<th><strong>TTT</strong></th>\n</tr>\n<tr>\n<td align=\"left\"><strong>Probability</strong></td>\n<td>$\\var{h}^2\\times\\var{t}=\\var{h^2*t}$</td>\n<td>$\\var{h}\\times\\var{t}^2=\\var{h*t^2}$</td>\n<td>$\\var{h}\\times\\var{t}^2=\\var{h*t^2}$</td>\n<td>$\\var{t}^3=\\var{t^3}$</td>\n</tr>\n<tr>\n<td align=\"left\"><strong>$X$</strong></td>\n<td>$2$</td>\n<td>$1$</td>\n<td>$1$</td>\n<td>$0$</td>\n</tr>\n</tbody>\n</table>\n<p>So we see that:</p>\n<p>$P(X=0)=\\var{h0},\\;P(X=1)=\\var{h^2*t}+3\\times \\var{h*t^2}=\\var{h1}$</p>\n<p>$P(X=2)=2\\times \\var{h^2*t}=\\var{h2},\\;P(X=3)=\\var{h3}$</p>\n<p>b)</p>\n<p>The expectation is given by:</p>\n<p>$\\operatorname{E}[X]=\\sum x \\;P(X=x)=0\\times \\var{h0}+1\\times \\var{h1}+2\\times \\var{h2}+3\\times \\var{h3}=\\var{ex}$ (exactly).</p>\n<p>c)</p>\n<p>For this discrete distribution we use the formula:</p>\n<p>\\[\\begin{eqnarray*}\\operatorname{Var}(X)&amp;=&amp;\\operatorname{E}[X^2]-\\operatorname{E}(X)^2\\\\&amp;=&amp;0\\times \\var{h0}+1\\times \\var{h1}+4\\times \\var{h2}+9\\times \\var{h3}-\\var{ex}^2\\\\&amp;=&amp;\\var{precround(var,4)}\\end{eqnarray*}\\] to 4 decimal places.</p>", "rulesets": {}, "extensions": [], "variables": {"var": {"name": "var", "group": "Ungrouped variables", "definition": "9*h3+4*h2+h1-ex^2", "description": "", "templateType": "anything"}, "h0": {"name": "h0", "group": "Ungrouped variables", "definition": "t^3", "description": "", "templateType": "anything"}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(0.2..0.9#0.05 except 0.5)", "description": "", "templateType": "anything"}, "h3": {"name": "h3", "group": "Ungrouped variables", "definition": "h^3", "description": "", "templateType": "anything"}, "thismany": {"name": "thismany", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "1-h", "description": "", "templateType": "anything"}, "h1": {"name": "h1", "group": "Ungrouped variables", "definition": "h^2*t+3*h*t^2", "description": "", "templateType": "anything"}, "h2": {"name": "h2", "group": "Ungrouped variables", "definition": "2*h^2*t", "description": "", "templateType": "anything"}, "ex": {"name": "ex", "group": "Ungrouped variables", "definition": "3*h3+2*h2+h1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["h2", "h", "h0", "h1", "thismany", "t", "h3", "var", "ex"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$X$ can take the values $0,\\;1,\\;2$ or $3$.</p>\n<p>Complete the tables below in order to find the PMF. You must input all numbers as exact decimals - no rounding or approximations.</p>\n<p>First, find the probability of each outcome on tossing the coin.</p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td align=\"left\"><strong>Event</strong></td>\n<th><strong>HHH</strong></th><th><strong>HHT</strong></th><th><strong>HTH</strong></th><th><strong>HTT</strong></th></tr>\n<tr>\n<td align=\"left\"><strong>Probability</strong></td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n<tr>\n<td align=\"left\"><strong>$X$</strong></td>\n<td>$3$</td>\n<td>$2$</td>\n<td>$1$</td>\n<td>$1$</td>\n</tr>\n</tbody>\n</table>\n<table border=\"2\">\n<tbody>\n<tr>\n<td align=\"left\"><strong>Event</strong></td>\n<th><strong>THH</strong></th><th><strong>THT</strong></th><th><strong>TTH</strong></th><th><strong>TTT</strong></th></tr>\n<tr>\n<td align=\"left\"><strong>Probability</strong></td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n</tr>\n<tr>\n<td 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"", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "h^2*t", "maxValue": "h^2*t", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.25, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "h*t^2", "maxValue": "h*t^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.25, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "t*h^2", "maxValue": "t*h^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.25, "scripts": {}, "customMarkingAlgorithm": "", 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"correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.25, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "h0", "maxValue": "h0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "scripts": {}, "customMarkingAlgorithm": "", 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an exact decimal.</em></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "expectation", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex", "maxValue": "ex", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": 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