// Numbas version: exam_results_page_options
{"name": "Transient probabilities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "name": "Transient probabilities", "variable_groups": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "rulesets": {}, "parts": [{"marks": 0, "gaps": [{"precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "showPrecisionHint": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "allowFractions": false, "scripts": {}, "minValue": "{d31}", "mustBeReduced": false, "strictPrecision": false, "precision": "3", "maxValue": "{d31}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": []}, {"precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "showPrecisionHint": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "allowFractions": false, "scripts": {}, "minValue": "{d32}", "mustBeReduced": false, "strictPrecision": false, "precision": "3", "maxValue": "{d32}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": []}, {"precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "showPrecisionHint": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "allowFractions": false, "scripts": {}, "minValue": "{d33}", "mustBeReduced": false, "strictPrecision": false, "precision": "3", "maxValue": "{d33}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": []}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "prompt": "<p>What are the state probabilities at the start of day 3?</p>\n<p>Meteor &nbsp; &nbsp;&nbsp;[[0]]</p>\n<p>3&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[1]]</p>\n<p>Vodafone&nbsp;[[2]]</p>", "variableReplacements": []}, {"marks": 0, "gaps": [{"precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "showPrecisionHint": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "allowFractions": false, "scripts": {}, "minValue": "{d41}", "mustBeReduced": false, "strictPrecision": false, "precision": "3", "maxValue": "{d41}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": []}, {"precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "showPrecisionHint": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "allowFractions": false, "scripts": {}, "minValue": "{d42}", "mustBeReduced": false, "strictPrecision": false, "precision": "3", "maxValue": "{d42}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": []}, {"precisionMessage": "You have not given your answer to the correct precision.", "precisionPartialCredit": 0, "showPrecisionHint": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "allowFractions": false, "scripts": {}, "minValue": "{d43}", "mustBeReduced": false, "strictPrecision": false, "precision": "3", "maxValue": "{d43}", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": []}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "prompt": "<p>What are the state probabilities at the start of day 4?</p>\n<p>Meteor&nbsp; &nbsp; &nbsp; &nbsp;[[0]]</p>\n<p>3&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[1]]</p>\n<p>Vodafone&nbsp;[[2]]</p>", "variableReplacements": []}], "functions": {}, "variables": {"d32": {"description": "", "templateType": "anything", "name": "d32", "definition": "{d21}*{b}+{d22}*{d}+{d23}*{f}", "group": "Ungrouped variables"}, "e1": {"description": "", "templateType": "randrange", "name": "e1", "definition": "random(0.01..0.06#1)", "group": "Ungrouped variables"}, "d22": {"description": "", "templateType": "anything", "name": "d22", "definition": "({b}+{d}+{f})/3", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "randrange", "name": "b", "definition": "random(0.02..0.08#1)", "group": "Ungrouped variables"}, "d41": {"description": "", "templateType": "anything", "name": "d41", "definition": "{d31}*{a}+{d32}*{c}+{d33}*{e1}", "group": "Ungrouped variables"}, "d31": {"description": "", "templateType": "anything", "name": "d31", "definition": "{d21}*{a}+{d22}*{c}+{d23}*{e1}", "group": "Ungrouped variables"}, "c": {"description": "", "templateType": "randrange", "name": "c", "definition": "random(0.04..0.11#1)", "group": "Ungrouped variables"}, "a": {"description": "", "templateType": "randrange", "name": "a", "definition": "random(0.8..0.9#1)", "group": "Ungrouped variables"}, "d": {"description": "", "templateType": "randrange", "name": "d", "definition": "random(0.81..0.88#1)", "group": "Ungrouped variables"}, "d33": {"description": "", "templateType": "anything", "name": "d33", "definition": "1-{d31}-{d32}", "group": "Ungrouped variables"}, "d43": {"description": "", "templateType": "anything", "name": "d43", "definition": "1-{d41}-{d42}", "group": "Ungrouped variables"}, "d23": {"description": "", "templateType": "anything", "name": "d23", "definition": "1-{d21}-{d22}", "group": "Ungrouped variables"}, "d21": {"description": "", "templateType": "anything", "name": "d21", "definition": "({a}+{c}+{e1})/3", "group": "Ungrouped variables"}, "f": {"description": "", "templateType": "randrange", "name": "f", "definition": "random(0.05..0.1#1)", "group": "Ungrouped variables"}, "d42": {"description": "", "templateType": "anything", "name": "d42", "definition": "{d31}*{b}+{d32}*{d}+{d33}*{f}", "group": "Ungrouped variables"}}, "statement": "<p>The customers of three Mobile phone operators were asked if they were happy with their current provider and if not to indicate which provider they would change to.</p>\n<p>The results&nbsp;were analysed and the transition probabilities were evaluated as shown&nbsp;below:</p>\n<table style=\"margin-left: 30px;\" border=\"black\">\n<tbody style=\"padding-left: 30px;\">\n<tr style=\"padding-left: 30px;\">\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"padding-left: 30px;\">Future</td>\n<td style=\"padding-left: 30px;\"></td>\n</tr>\n<tr style=\"padding-left: 30px;\">\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"border-color: black;\">Meteor</td>\n<td style=\"border-color: black;\">3</td>\n<td style=\"border-color: black;\">Vodafone</td>\n</tr>\n<tr style=\"padding-left: 30px;\">\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"padding-left: 30px;\">Meteor</td>\n<td style=\"border-color: black;\">\\(\\var{a}\\)</td>\n<td style=\"border-color: black;\">\\(\\var{b}\\)</td>\n<td style=\"border-color: black;\">\\(\\simplify{1-{a}-{b}}\\)</td>\n</tr>\n<tr style=\"padding-left: 30px;\">\n<td style=\"padding-left: 30px;\">Present</td>\n<td style=\"padding-left: 30px;\">3</td>\n<td style=\"border-color: black;\">\\(\\var{c}\\)</td>\n<td style=\"border-color: black;\">\\(\\var{d}\\)</td>\n<td style=\"border-color: black;\">\\(\\simplify{1-{c}-{d}}\\)</td>\n</tr>\n<tr style=\"padding-left: 30px;\">\n<td style=\"padding-left: 30px;\"></td>\n<td style=\"padding-left: 30px;\">Vodafone</td>\n<td style=\"border-color: black;\">\\(\\var{e1}\\)</td>\n<td style=\"border-color: black;\">\\(\\var{f}\\)</td>\n<td style=\"border-color: black;\">\\(\\simplify{1-{e1}-{f}}\\)</td>\n</tr>\n</tbody>\n</table>\n<p>Assume each of the operators&nbsp;initially had equal market share.</p>", "preamble": {"css": "", "js": ""}, "extensions": [], "ungrouped_variables": ["a", "b", "c", "d", "e1", "f", "d21", "d22", "d23", "d31", "d32", "d33", "d41", "d42", "d43"], "advice": "<p>The matrix of transition probabilities is given by:</p>\n<p style=\"padding-left: 30px;\">\\(\\mathbf{A}=\\begin{pmatrix} \\var{a}&amp;\\var{b}&amp;\\simplify{1-{a}-{b}}\\\\ \\var{c}&amp;\\var{d}&amp;\\simplify{1-{c}-{d}}\\\\\\var{e1}&amp;\\var{f}&amp;\\simplify{1-{e1}-{f}} \\end{pmatrix}\\)</p>\n<p>\\(\\Pi(n)\\) represents the proportions in each state after \\(n\\) time periods have elapsed.</p>\n<p>Recall&nbsp; \\(\\Pi(n)=\\Pi(0)A^n\\)</p>\n<p>The initial state probabilities are: \\(\\Pi(0)=\\left(\\frac{1}{3}\\,\\,\\frac{1}{3}\\,\\,\\frac{1}{3}\\right)\\)</p>\n<p></p>\n<p>The start of day 3 is after 2 days completed i.e. \\(\\Pi(2)\\)&nbsp; and the start of day 4 is after 3 days completed i.e. \\(\\Pi(3)\\)</p>\n<p></p>\n<p style=\"padding-left: 30px;\">\\(\\Pi(1)=\\Pi(0)A=\\left(\\frac{1}{3}\\,\\,\\frac{1}{3}\\,\\,\\frac{1}{3}\\right)\\begin{pmatrix} \\var{a}&amp;\\var{b}&amp;\\simplify{1-{a}-{b}}\\\\ \\var{c}&amp;\\var{d}&amp;\\simplify{1-{c}-{d}}\\\\\\var{e1}&amp;\\var{f}&amp;\\simplify{1-{e1}-{f}} \\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\Pi(1)=\\left(\\var{d21}\\,\\,\\var{d22}\\,\\,\\var{d23}\\right)\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\">\\(\\Pi(2)=\\Pi(1)A=\\left(\\var{d21}\\,\\,\\var{d22}\\,\\,\\var{d23}\\right)\\begin{pmatrix} \\var{a}&amp;\\var{b}&amp;\\simplify{1-{a}-{b}}\\\\ \\var{c}&amp;\\var{d}&amp;\\simplify{1-{c}-{d}}\\\\\\var{e1}&amp;\\var{f}&amp;\\simplify{1-{e1}-{f}} \\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\Pi(2)=\\left(\\var{d31}\\,\\,\\var{d32}\\,\\,\\var{d33}\\right)\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\">\\(\\Pi(3)=\\Pi(2)A=\\left(\\var{d31}\\,\\,\\var{d32}\\,\\,\\var{d33}\\right)\\begin{pmatrix} \\var{a}&amp;\\var{b}&amp;\\simplify{1-{a}-{b}}\\\\ \\var{c}&amp;\\var{d}&amp;\\simplify{1-{c}-{d}}\\\\\\var{e1}&amp;\\var{f}&amp;\\simplify{1-{e1}-{f}} \\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\Pi(3)=\\left(\\var{d41}\\,\\,\\var{d42}\\,\\,\\var{d43}\\right)\\)</p>", "tags": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}