We have:
\n\\[\\begin{eqnarray*} P(X \\gt \\var{c})&=&\\int_{\\var{c}}^{\\var{b}}f(x)\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\int_{\\var{c}}^{\\var{b}}\\simplify[std]{({f1}-{s1}x)}\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\left(\\simplify[std,!otherNumbers]{{f1}({b}-{c})-{s1}({b}^2-{c}^2)/2}\\right)\\\\ &=&\\simplify[std]{{2*f1 * (b -c) + s1 * (c ^ 2 -(b ^ 2))} / {2*n1} }\\\\ &=& \\var{pc} \\end{eqnarray*} \\] to 4 decimal places.
\n\\[\\begin{eqnarray*} P(X \\gt \\var{d} | X \\gt \\var{c})&=&\\frac{P(X \\gt \\var{d}\\;\\;\\textrm{and}\\;\\; X \\gt \\var{c})}{P(X \\gt \\var{c})}\\\\ &=&\\frac{P(X \\gt \\var{d})}{P(X \\gt \\var{c})} \\end{eqnarray*} \\] as $\\var{d} \\gt \\var{c}$.
\nBut,
\\[\\begin{eqnarray*} P(X \\gt \\var{d})&=&\\int_{\\var{d}}^{\\var{b}}f(x)\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\int_{\\var{d}}^{\\var{b}}\\simplify[std]{({f1}-{s1}x)}\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\left(\\simplify[std,!otherNumbers]{{f1}({b}-{d})-{s1}({b}^2-{d}^2)/2}\\right)\\\\ &=&\\simplify[std]{{2*f1 * (b -d) + s1 * (d ^ 2 -(b ^ 2))} / {2*n1} }\\\\ &=& \\var{pd1} \\end{eqnarray*} \\] to 4 decimal places.
Hence \\[\\begin{eqnarray*} P(X \\gt \\var{d} | X \\gt \\var{c})&=&\\frac{P(X \\gt \\var{d})}{P(X \\gt \\var{c})}\\\\ &=&\\frac{\\var{pd1}}{\\var{pc}}\\\\ &=&\\var{pd} \\end{eqnarray*} \\] to 2 decimal places.
", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n \n \n$P(X \\gt \\var{c})=\\;\\;$[[0]]
\n \n \n \nInput to 4 decimal places.
\n \n \n ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{pc+tol}", "minValue": "{pc-tol}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n$P(X \\gt \\var{d} | X \\gt \\var{c})=\\;\\;$[[0]]
\nInput to 2 decimal places.
\n ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{pd+tol1}", "minValue": "{pd-tol1}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\nGiven the pdf
\n\\[\\begin{eqnarray*} f(x)&=&\\simplify[std]{({f1}-{s1}x)/{n1}}\\;\\;\\var{a} \\leq x \\leq \\var{b}\\\\ f(x)&=&0\\;\\;\\;\\textrm{otherwise} \\end{eqnarray*} \\]
find the following probabilities:
8/07/2012:
\n \t\tAdded tags.
\n \t\tSet new tolerance variables, tol=0 for first question and tol1=0 for second question. Aslo included statement that second question is to be entered to 2 dps.
\n \t\tThere is an image to be included in the Advice. This needs to be done.
\n \t\tChecked calculations, OK.
\n \t\t23/07/2012:
\n \t\tAdded description.
\n \t\t1/08/2012:
\n \t\tQuestion appears to be working correctly.
\n \t\t21/12/2012:
\n \t\tChecked calculation. Added tag tested1.
\n \t\tAdded query and diagram tags re possible inclusion of a diagram - which could be dynamic?
\n \t\tChecked rounding, OK. Added tag cr1.
\n \t\t", "description": "Given the pdf $f(x)=\\frac{a-bx}{c},\\;r \\leq x \\leq s,\\;f(x)=0$ else, find $P(X \\gt p)$, $P(X \\gt q | X \\gt t)$.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}