Normal Distribution, Binomial Distribution, Poisson Distribution
\nrebel
\nrebelmaths
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", "licence": "None specified"}, "statement": "\nWhich of the following random variables could be modelled with a binomial distribution and which could be modelled with a Poisson distribution?
\nYou will lose 1 mark for every incorrect answer. The minimum mark is 0.
\n ", "advice": "No solution given.
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If a random sample of eight CIT students is chosen, calculate the probability that...
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Please give your answer to at least 3 decimal places.
\nIt is estimated that $\\var{p_perc}$% of all CIT students cycle to college. A random sample of $\\var{n}$ CIT students is chosen.
\n", "advice": "Part (a)
\nIf a random variable $X$ follows a binomial distribution with parameters $n$ and $p$. The probability of $r$ successes out of $n$ trials is given by:
\n$P(X=r)=P(r,n)=C^n_{r}p^{r}q^{n-r}$
\nwhere $p$ is the probability of success for each trial and $q$ is the probability of failure for each trial.
\nThe probability that a student cycles to college is $\\var{p}$, therefore $p=\\var{p}$ and $q=1-\\var{p}=\\var{q}$.
\nWe are interested in claculating the probability that none of the sample of $\\var{n}$ students cycle to college so $r=0$ and $n=\\var{n}$
\n$P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}$
\n$P(\\var{r0}, \\var{n})= \\var{pr0}$
\n\n
Part (b)
\nWe are interested in claculating the probability that at least $\\var{r}$ of the $\\var{n}$ students cycle to college. Let $X$ represent the number of students that cycle to college. We need to calculate:
\n$P(X \\geq \\var{r}) = P(X= \\var{r}) + P(X= \\var{r+1})+...+ P(X=\\var{n})$
\n\n
Since $P(X=\\var{r0})+P(X=\\var{r0+1})+...+P(X=\\var{n})=\\var{r0+1}$
\nWe may write
\n$P(X \\geq \\var{r}) = 1-P(X= \\var{r0}) - P(X=\\var{r0+1})-...- P(X=\\var{r-1})$
\n\n
where
\n$P(X= \\var{r0})=P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}=\\var{pr0}$
\n$P(X=1) =P(1, \\var{n})= C^\\var{n}_{1}$ $\\var{p}^{1}$ $\\var{q}^{\\var{n}-1}$ $=\\var{pr1}$
\n$P(X=2) = P(2, \\var{n})=$ $C^\\var{n}_{2}$ $\\var{p}^{2}$ $\\var{q}^{\\var{n}-2}$ $=\\var{pr2}$
\n\nThen
\n$P(X \\geq \\var{r}) = 1-\\var{qn}-\\var{pr1}-\\var{pr2}=\\var{answer2}$
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a Poisson distribution, with a mean of four patients per hour.
Calculate the probability that in a particular one-hour period
Please give your answer to at least 3 decimal places.
\nThe number of patients arriving at a dentist’s surgery each afternoon follows a Poisson distribution, with a mean of $\\var{l}$ patients per hour.
Part (a)
\nRemember that for a Poisson random variable:
\\begin{align}
\\operatorname{P}(X=x)&=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\
\\end{align}
1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{x}) &=& \\frac{\\var{l} ^ {\\var{x}}e ^ { -\\var{l}}} {\\var{x}!}\\\\& =& \\var{answer1} \\end{eqnarray*} \\] to 3 decimal places.
\n\n
Part (b)
\nThe probability that in a particular one hour period, the number of patients entering the waiting room will be between $\\var{x}$ and $\\var{y}$ inclusive is given by:
\n$P(\\var{x} \\leq X\\leq\\var{y}) = P(X=\\var{x}) + P(X=\\var{x+1}) +P(X=\\var{y})$
\nwhere
\n$P(X=\\var{x}) =\\frac{\\var{l}^{\\var{x}}e^{-\\var{l}}}{\\var{x}!}=\\var{prx}$
\n$P(X=\\var{x+1}) =\\frac{\\var{l}^{\\var{x+1}}e^{-\\var{l}}}{\\var{x+1}!}=\\var{prx1}$
\n$P(X=\\var{y}) =\\frac{\\var{l}^{\\var{y}}e^{-\\var{l}}}{\\var{y}!}=\\var{pry}$
\nHence
\n$P(\\var{x} \\leq X \\leq \\var{y})=$ $\\var{prx}+\\var{prx1}+\\var{pry}=\\var{answer2}$
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\nFinding probabilities using the Poisson distribution.
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{pre} $\\var{thismany}$.
\n{something} $\\var{number1}$ {else}
\nThe Poisson distribution formula: $P(r)=\\frac{\\lambda^re^{-\\lambda}}{r!}$ or $P(r)=e^{-\\lambda}\\left[\\frac{\\lambda^r}{r!}\\right]$
", "advice": "a)
\n1. $X \\sim \\operatorname{Poisson}(\\var{thismany})$, so $\\lambda = \\var{thismany}$.
\n\nb)
\nRemember that for a Poisson random variable:
\\begin{align}
\\operatorname{P}(X=x)&=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\
\\end{align}
1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\frac{\\var{thismany} ^ {\\var{thisnumber}}e ^ { -\\var{thismany}}} {\\var{thisnumber}!}\\\\& =& \\var{prob1} \\end{eqnarray*} \\] to 3 decimal places.
\n\n
2. If an employee receives a warning then he or she must have sold less than {number1}.
\nHence we need to find :
\n\\[ \\begin{eqnarray*}\\operatorname{P}(X < \\var{number1})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\& =& \\simplify[all,!collectNumbers]{e ^ { -thismany} + {thismany} * e ^ { -thismany} + {v} * (({thismany} ^ 2 * e ^ { -thismany}) / 2)} \\\\&=& \\var{prob2} \\end{eqnarray*} \\]
\nto 3 decimal places.
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Assuming a Poisson distribution for {descX}, write down the value of $\\lambda$.
\n$\\lambda = $[[0]]?
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\n$\\operatorname{P}(r=\\var{thisnumber})=$? [[0]] (to 3 decimal places).
\n\n
Find the probability that {thisaswell}
\n$\\operatorname{P}(r < \\var{number1})= $? [[1]] (to 3 decimal places).
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob1-tol", "maxValue": "prob1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob2-0.005", "maxValue": "prob2+0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Clodagh's copy of Poisson Distribution (printing errors)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": ["rebelmaths"], "metadata": {"description": "rebelmaths
\nPrinting errors in the work produced by a particular film occur randomly at an average rate of p per page.
\n
i.What is the probability that a one page document will contain x1 printing error(s)?
ii.If a n page document is printed, calculate the probability of having more than x2 errors. Assume a Poisson distribution.
Please give your answer to at least 3 decimal places.
\nPrinting errors in the work produced by a particular film occur randomly at an average rate of $\\var{l}$ per page.
\n\n
The Poisson distribution formula: $P(r)=\\frac{\\lambda^re^{-\\lambda}}{r!}$ or $P(r)=e^{-\\lambda}\\left[\\frac{\\lambda^r}{r!}\\right]$
", "advice": "Part (a)
\nRemember that for a Poisson random variable:
\\begin{align}
\\operatorname{P}(X=x)&=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\
\\end{align}
1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{x}) &=& \\frac{\\var{l} ^ {\\var{x}}e ^ { -\\var{l}}} {\\var{x}!}\\\\& =& \\var{answer1} \\end{eqnarray*} \\] to 3 decimal places.
\n\n
Part (b)
\n2. For a $\\var{n}$ page document, $\\lambda=\\var{n} \\times \\var{l} = \\var{l2}$
\nThe probability of having less than $\\var{y}$ errors is given by:
\n$P(X < \\var{y}) = P(X=0) + P(X=1) +P(X=2)$
\nwhere
\n$P(X=0) =\\frac{\\var{l2}^{0}e^{-\\var{l2}}}{0!}=\\var{pr0}$
\n$P(X=1) =\\frac{\\var{l2}^{1}e^{-\\var{l2}}}{1!}=\\var{pr1}$
\n$P(X=1) =\\frac{\\var{l2}^{2}e^{-\\var{l2}}}{2!}=\\var{pr2}$
\nHence
\n$P(X < \\var{y})$ = $\\var{pr0}+\\var{pr1}+\\var{pr2}=\\var{answer2}$
", "rulesets": {}, "variables": {"answer1": {"name": "answer1", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!", "description": "", "templateType": "anything"}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "random(0.05..0.4#0.1)", "description": "average, lambda
", "templateType": "anything"}, "answer2": {"name": "answer2", "group": "Ungrouped variables", "definition": "if(y=3, ((e^-l2)*(l2^0))/0!+((e^-l2)*(l2^(1)))/(1)!+((e^-l2)*(l2^2))/2!,((e^-l2)*(l2^0))/0!+((e^-l2)*(l2^(1)))/(1)!)", "description": "", "templateType": "anything"}, "l2": {"name": "l2", "group": "Ungrouped variables", "definition": "l*n", "description": "", "templateType": "anything"}, "pr0": {"name": "pr0", "group": "Ungrouped variables", "definition": "((e^-l2)*(l2^0))/0!", "description": "", "templateType": "anything"}, "pr2": {"name": "pr2", "group": "Ungrouped variables", "definition": "((e^-l2)*(l2^2))/2!", "description": "", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "3", "description": "upper value of X
", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "number of customers entering the shop
", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..15#1)", "description": "time interval
", "templateType": "anything"}, "pr1": {"name": "pr1", "group": "Ungrouped variables", "definition": "((e^-l2)*(l2^1))/1!", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["l", "x", "n", "y", "answer1", "answer2", "l2", "pr0", "pr1", "pr2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is the probability that a one page document will contain exactly $\\var{x}$ printing error(s)?
", "minValue": "answer1-0.001", "maxValue": "answer1+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "If a $\\var{n}$ page document is printed, calculate the probability of having less than $\\var{y}$ errors.
", "minValue": "answer2 -0.001", "maxValue": "answer2 +0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Clodagh's copy of BS3.3 Binomial Cookies", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": ["Binomial Distribution", "binomial distribution", "Binomial distribution", "expectation", "expected number", "probabilities", "probability", "Probability", "rebelmaths", "sc", "standard deviation", "statistical distributions", "statistics"], "metadata": {"description": "rebelmaths
\nApplication of the binomial distribution given probabilities of success of an event.
\nFinding probabilities using the binomial distribution.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{pre} $\\var{thismany}${post}
\n{something} $\\var{number1}$ {else}
\n", "advice": "
a)
\n$X \\sim \\operatorname{bin}(\\var{number1},\\var{prob})$, so $n= \\var{number1},\\;\\;p=\\var{prob}$.
\n\nb)
\n1. \\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\dbinom{\\var{number1}}{\\var{thisnumber}}\\times\\var{prob}^{\\var{thisnumber}}\\times(1-\\var{prob})^{\\var{number1-thisnumber}}\\\\& =& \\var{comb(number1,thisnumber)} \\times\\var{prob}^{\\var{thisnumber}}\\times\\var{1-prob}^{\\var{number1-thisnumber}}\\\\&=&\\var{prob1}\\end{eqnarray*} \\] to 3 decimal places.
\n\n
2.
\n\\[ \\begin{eqnarray*}\\operatorname{P}(X \\leq \\var{thatnumber})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)+ {v1}*P(X = 3)}\\\\& =& \\simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})+ {v1} * ({number1} * {number1 -1}*{number1-2}/(3*2))* {prob} ^ 3 *( {1 -prob} ^ {number1 -3})}\\\\& =& \\var{prob3}\\end{eqnarray*} \\]
\nto 3 decimal places.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pre": {"name": "pre", "group": "Ungrouped variables", "definition": "' '", "description": "", "templateType": "anything", "can_override": false}, "descx1": {"name": "descx1", "group": "Ungrouped variables", "definition": "\"number of chocolate chip cookies in our sample:\"", "description": "", "templateType": "anything", "can_override": false}, "tprob3c": {"name": "tprob3c", "group": "Ungrouped variables", "definition": "if(thatnumber=1,(1-prob)^number1+number1*prob*(1-prob)^(number1-1),0)", "description": "", "templateType": "anything", "can_override": false}, "tprob3b": {"name": "tprob3b", "group": "Ungrouped variables", "definition": "if(thatnumber=3,(1-prob)^number1+number1*prob*(1-prob)^(number1-1)+number1*(number1-1)*prob^2*(1-prob)^(number1-2)/2+number1*(number1-1)*(number1-2)*prob^3*(1-prob)^(number1-3)/(3*2),0)", "description": "", "templateType": "anything", 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"templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.001", "description": "", "templateType": "anything", "can_override": false}, "prob": {"name": "prob", "group": "Ungrouped variables", "definition": "thismany/100", "description": "", "templateType": "anything", "can_override": false}, "thisaswell": {"name": "thisaswell", "group": "Ungrouped variables", "definition": "\"our selection contains no more than \"", "description": "", "templateType": "anything", "can_override": false}, "else": {"name": "else", "group": "Ungrouped variables", "definition": "\"biscuits are selected at random.\"", "description": "", "templateType": "anything", "can_override": false}, "v1": {"name": "v1", "group": "Ungrouped variables", "definition": "if(thatnumber=3,1,0)", "description": "", "templateType": "anything", "can_override": false}, "number1": {"name": "number1", "group": "Ungrouped variables", "definition": "random(5..12)*random([1,1,1,1,1,1,2,2,5])", "description": "", "templateType": "anything", "can_override": false}, "post": {"name": "post", "group": "Ungrouped variables", "definition": "\"% of biscuits made by a baker are chocolate chip cookies.\"", "description": "", "templateType": "anything", "can_override": false}, "prob2": {"name": "prob2", "group": "Ungrouped variables", "definition": "precround(tprob2,3)", "description": "", "templateType": "anything", "can_override": false}, "prob3": {"name": "prob3", "group": "Ungrouped variables", "definition": "precround(tprob3,3)", "description": "", "templateType": "anything", "can_override": false}, "prob1": {"name": "prob1", "group": "Ungrouped variables", "definition": "precround(tprob1,3)", "description": "", "templateType": "anything", "can_override": false}, "thatnumber": {"name": "thatnumber", "group": "Ungrouped variables", "definition": "random(1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "thismany": {"name": "thismany", "group": "Ungrouped variables", "definition": "random(8..43)", "description": "", "templateType": "anything", "can_override": false}, "this": {"name": "this", "group": "Ungrouped variables", "definition": "\"our selection contains exactly \"", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(thatnumber=1,0,1)", "description": "", "templateType": "anything", "can_override": false}, "tprob1": {"name": "tprob1", "group": "Ungrouped variables", "definition": "comb(number1,thisnumber)*prob^thisnumber*(1-prob)^(number1-thisnumber)", "description": "", "templateType": "anything", "can_override": false}, "tprob3": {"name": "tprob3", "group": "Ungrouped variables", "definition": "tprob3a+tprob3b+tprob3c", "description": "", "templateType": "anything", "can_override": false}, "tprob2": {"name": "tprob2", "group": "Ungrouped variables", "definition": 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Assuming a binomial distribution for {descX}, write down the values of $n$ and $p$.
\n$n=\\; $[[0]] $p=\\;$[[1]]
\n\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "number1", "maxValue": "number1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob", "maxValue": "prob", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "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": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the probability that {this} $\\var{thisnumber}$ {things}
\n$\\operatorname{P}(r=\\var{thisnumber})=$ [[0]] (to 3 decimal places).
\n\n
Find the probability that {thisaswell} {thatnumber} {things}
\n$\\operatorname{P}(r\\leq\\var{thatnumber})=$ [[1]] (to 3 decimal places).
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob1-tol", "maxValue": "prob1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob3-tol", "maxValue": "prob3+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Binomial (practice of formula)", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": ["binomial distribution", "Binomial Distribution", "Binomial distribution", "CDF", "cdf", "CDF of binomial distribution", "cr1", "cumulative density function", "Discrete random variables.", "distributions", "Expectation of binomial distribution", "probability", "Probability", "random variables", "REBEL", "rebel", "Rebel", "rebelmaths", "statistics", "tested1", "variance of binomial distribution"], "metadata": {"description": "$X \\sim \\operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \\leq b)$, $E[X],\\;\\operatorname{Var}(X)$.
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Answer the following questions on the Binomial Distribution.
\nSuppose \\[X \\sim \\operatorname{Binomial}(\\var{n},\\var{p}),\\]
\nthat is $n=\\var{n}$ and $p=\\var{p}$.
", "advice": "a)
\\[\\simplify[std,!otherNumbers]{P(X = {x1}) = {n}! / ({n -x1}! * {x1}!) * {p} ^ {x1} * (1 -{p}) ^ {n -x1}} = \\var{ans1}\\]
to 3 decimal places.
\nb)
\nWe have:
\n\\[ \\begin{eqnarray*} F_X (\\var{x2}) &=& P(X \\le \\var{x2}) =\\simplify[std]{ P(X = 0) + P(X = 1) + P(X = 2) + {v3} * P(X = 3) + {v4} * P(X = 4)}\\\\ &=& \\simplify[unitFactor,zeroTerm,zeroFactor]{(1 -{p}) ^ {n} + {n} * (1 -{p}) ^ {n -1} * {p} + {(n * (n -1)) / 2} * (1 -{p}) ^ {n -2} * {p} ^ 2 + {v3} * {Comb(n , 3)} * (1 -{p}) ^ {n -3} * {p} ^ 3 + {v4} * {Comb(n , 4)} * (1 -{p}) ^ {n -4} * {p} ^ 4}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\]
to 3 decimal places.
Compute $P(r=\\var{x1})=\\;\\;$[[0]] (to 3 decimal places).
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", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{ans2-tol}", "maxValue": "{ans2+tol}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "type": "question"}, {"name": "Probabilities from a normal distribution - Electrician", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}], "tags": ["checked2015", "MAS1403", "rebel", "REBEL", "Rebel", "rebelmaths"], "metadata": {"description": "Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "The salary of an irish electrician is normally distributed with mean €{m} and standard deviation €{s}.
\n", "advice": "
1. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s})} = P(Z>\\var{zupper}) = 1-P(Z<\\var{zupper})=1-\\var{p1} = \\var{prob2}$ to 2 decimal places.
\n2.
\n$\\simplify[all,!collectNumbers]{P({lower} < X < {upper}) = P(X < {upper})-P(X < {lower})}=P(Z<\\var{zupper})-P(Z<-\\var{zlower}) =\\var{p1}-\\var{p2} = \\var{prob3}$ to 2 decimal places.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"zupper": {"name": "zupper", "group": "Ungrouped variables", "definition": "precround((upper-m)/s,2)", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "1-p", "description": "", "templateType": "anything", "can_override": false}, "amount": {"name": "amount", "group": "Ungrouped variables", "definition": "\"electricity consumption\"", "description": "", "templateType": "anything", "can_override": false}, "units1": {"name": "units1", "group": "Ungrouped variables", "definition": "\"k Wh\"", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "random(m+0.5s..m+1.5*s#10000)", "description": "", "templateType": "anything", "can_override": false}, "prob3": {"name": "prob3", "group": "Ungrouped variables", "definition": "precround(p1-p2,2)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(5000..15000#1000)", "description": "", "templateType": "anything", "can_override": false}, "prob1": {"name": "prob1", "group": "Ungrouped variables", "definition": "precround(1-p,2)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(30000..50000#2000)", "description": "", "templateType": "anything", "can_override": false}, "prob2": {"name": "prob2", "group": "Ungrouped variables", "definition": "precround(1-p1,2)", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.01", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "random(m-1.5*s..m-0.5s#10000)", "description": "", "templateType": "anything", "can_override": false}, "stuff": {"name": "stuff", "group": "Ungrouped variables", "definition": "\"a frozen foods warehouse each week in the summer months \"", "description": "", "templateType": "anything", "can_override": false}, "zlower": {"name": "zlower", "group": "Ungrouped variables", "definition": "precround((m-lower)/s,2)", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "precround(normalcdf(zupper,0,1),4)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "precround(normalcdf(zlower,0,1),4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "s", "zupper", "p", "amount", "p2", "tol", "zlower", "stuff", "prob2", "prob3", "prob1"], "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": "If one electrician is chosen at random, what is probability that this electrician earns over €{upper}?
\nProbability = [[0]](to 2 decimal places)
\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob2-tol", "maxValue": "prob2+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "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": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "If one electrician is chosen at random, what is probability that this electrician earns less than €{lower} ?
\nProbability = [[0]](to 2 decimal places)
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob1-tol", "maxValue": "prob1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "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": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "If one electrician is chosen at random, what is probability that this electrician earns between €{lower} and €{upper}?
\nProbability = [[0]](to 2 decimal places)
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob3-tol", "maxValue": "prob3+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Clodagh's copy of Julie's copy of Calculate probabilities from normal distribution, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}], "functions": {}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "amount", "zupper", "p", "s", "stuff", "tol", "zlower", "prob2", "prob1"], "tags": ["ACC1012", "checked2015", "MAS1403", "rebelmaths"], "advice": "1. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X < {lower}) = P(Z < ({lower} -{m}) / {s}) =1 -P(Z < {m-lower}/{s})} = 1-P(z<\\var{zlower})=1 -\\var{p} = \\var{prob1}$ to 2 decimal places.
\n2. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s}) = 1 -P(Z < {upper-m}/{s})} = 1-P(z<\\var{zupper})=1-\\var{p1} = \\var{prob2}$ to 2 decimal places.
", "rulesets": {}, "parts": [{"prompt": "Find the probability that in a particular week the {amount} is less than {lower} {units1}:
\nProbability = ?[[0]](to 2 decimal places)
\nFind the probability that in a particular week the {amount} is greater than {upper} {units1}:
\nProbability = ?[[1]](to 2 decimal places)
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "prob1+tol", "minValue": "prob1-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "prob2+tol", "minValue": "prob2-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "The {amount}, $X$, of {stuff} is normally distributed with mean {m}k Wh and standard deviation {s}{units1}.
\n\n", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"units1": {"definition": "\"k Wh\"", "templateType": "anything", "group": "Ungrouped variables", "name": "units1", "description": ""}, "upper": {"definition": "random(m+0.5s..m+1.5*s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper", "description": ""}, "lower": {"definition": "random(m-1.5*s..m-0.5s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "p1": {"definition": "precround(normalcdf(zupper,0,1),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "m": {"definition": "random(750..1250#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "amount": {"definition": "\"electricity consumption\"", "templateType": "anything", "group": "Ungrouped variables", "name": "amount", "description": ""}, "zupper": {"definition": "precround((upper-m)/s,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "zupper", "description": ""}, "p": {"definition": "precround(normalcdf(zlower,0,1),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "s": {"definition": "random(60..100#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "stuff": {"definition": "\"a frozen foods warehouse each week in the summer months \"", "templateType": "anything", "group": "Ungrouped variables", "name": "stuff", "description": ""}, "tol": {"definition": "0.01", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "zlower": {"definition": "precround((m-lower)/s,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "zlower", "description": ""}, "prob2": {"definition": "precround(1-p1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob1": {"definition": "precround(1-p,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}}, "metadata": {"description": "
rebelmaths
\nGiven a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Clodagh's copy of Calculate probabilities from a normal distribution", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}], "functions": {}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "s", "zupper", "p", "amount", "p2", "tol", "zlower", "stuff", "prob2", "prob3", "prob1"], "tags": ["checked2015", "MAS1403", "rebelmaths"], "advice": "1. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X < {lower}) = P(Z < ({lower} -{m}) / {s})} = P(Z<-\\var{zlower})= \\var{prob1}$ to 2 decimal places.
\n2. Converting to $\\operatorname{N}(0,1)$
\n$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s})} = P(Z>\\var{zupper}) = 1-P(Z<\\var{zupper})=1-\\var{p1} = \\var{prob2}$ to 2 decimal places.
\n3.
\n$\\simplify[all,!collectNumbers]{P({lower} < X < {upper}) = P(X < {upper})-P(X < {lower})}=P(Z<\\var{zupper})-P(Z<-\\var{zlower}) =\\var{p1}-\\var{p2} = \\var{prob3}$ to 2 decimal places.
", "rulesets": {}, "parts": [{"prompt": "Find the probability that in a particular week the {amount} is less than {lower} {units1}:
\nProbability = ?[[0]](to 2 decimal places)
\nFind the probability that in a particular week the {amount} is greater than {upper} {units1}:
\nProbability = ?[[1]](to 2 decimal places)
\nFind the probability that in a particular week the {amount} is between {lower}{units1} and {upper} {units1}:
\nProbability = ?[[2]](to 2 decimal places)
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "prob1+tol", "minValue": "prob1-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "prob2+tol", "minValue": "prob2-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "prob3+tol", "minValue": "prob3-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "2", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "The {amount}, $X$, of {stuff} is normally distributed with mean {m}{units1} and standard deviation {s}{units1}.
\n", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"units1": {"definition": "\"k Wh\"", "templateType": "anything", "group": "Ungrouped variables", "name": "units1", "description": ""}, "upper": {"definition": "random(m+0.5s..m+1.5*s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper", "description": ""}, "lower": {"definition": "random(m-1.5*s..m-0.5s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "p1": {"definition": "precround(normalcdf(zupper,0,1),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "m": {"definition": "random(750..1250#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "amount": {"definition": "\"electricity consumption\"", "templateType": "anything", "group": "Ungrouped variables", "name": "amount", "description": ""}, "zupper": {"definition": "precround((upper-m)/s,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "zupper", "description": ""}, "p": {"definition": "precround(normalcdf(zlower,0,1),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "s": {"definition": "random(60..100#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "p2": {"definition": "1-p", "templateType": "anything", "group": "Ungrouped variables", "name": "p2", "description": ""}, "tol": {"definition": "0.01", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "zlower": {"definition": "precround((m-lower)/s,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "zlower", "description": ""}, "stuff": {"definition": "\"a frozen foods warehouse each week in the summer months \"", "templateType": "anything", "group": "Ungrouped variables", "name": "stuff", "description": ""}, "prob2": {"definition": "precround(1-p1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob3": {"definition": "precround(p1-p2,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob3", "description": ""}, "prob1": {"definition": "precround(1-p,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}}, "metadata": {"description": "
rebelmaths
\nGiven a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}