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Merge pull request #1183 from lahvak/UBC_ECON_percent_fix
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Fix the percentages in the problem text
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@drdrew42
drdrew42 authored Dec 19, 2024
2 parents d1aae08 + bf73ccd commit a185d81
Showing 1 changed file with 68 additions and 103 deletions.
171 changes: 68 additions & 103 deletions OpenProblemLibrary/UBC/ECON/ECON325/hw03/hw03_q12.pg
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Original file line number Diff line number Diff line change
Expand Up @@ -15,12 +15,9 @@ DOCUMENT();
## Initializations: (Required)
loadMacros(
"PGstandard.pl",
"PGchoicemacros.pl",
"parserRadioButtons.pl",
"MathObjects.pl",
"parserMultiAnswer.pl",
"RserveClient.pl",
"answerHints.pl",
"PGML.pl",
"PGcourse.pl"
);

Expand Down Expand Up @@ -83,114 +80,82 @@ q2<-p2*pt2/(p1*pt1 + p2*pt2 + p3*pt3)
round(q2, 2)
');

# calculate percentages from probabilities

$p1perc = 100*$p1[0];
$p2perc = 100*$p2[0];
$p3perc = 100*$p3[0];

$q1ans = $q1[0];
$q2ans = $q2[0];
$pt1perc = 100*$pt1[0];
$pt2perc = 100*$pt2[0];
$pt3perc = 100*$pt3[0];


$q1ans = Compute("$q1[0]");
$q2ans = Compute("$q2[0]");


#########################################################
## Main Text: where all text goes (Required)
Context()->texStrings;
BEGIN_TEXT
The members of a consulting firm rent cars from three rental agencies. It is estimated that $p1[0] percent of cars come from agency 1, $p2[0] percent
of cars come from agency 2, and $p3[0] percent of cars come from agency 3.
It is also estimated that $pt1[0] percent of cars from agency 1 need a tune-up, $pt2[0] percent of cars from agency 2 need a tune-up, and $pt3[0] percent
of cars from agency 3 need a tune-up. Answer the following questions, rounding your answers to two decimal places where appropriate.

$BR
$BR
$BBOLD(a) $EBOLD What is the probability that a rental car delivered to the firm will need a tune-up?
\{ans_rule(7) \}
$BR
$BR
$BBOLD(b) $EBOLD If a rental car delivered to the firm needs a tune-up, what is the
probability that it came from agency 2? \{ans_rule(7) \}
END_TEXT

#########################################################
## Answers evaluation (Required)
ANS( num_cmp($q1ans, tol=>0.01) );
ANS( num_cmp($q2ans, tol=>0.01) );

BEGIN_PGML
The members of a consulting firm rent cars from three rental agencies.
It is estimated that [$p1perc] percent of cars come from agency 1, [$p2perc] percent
of cars come from agency 2, and [$p3perc] percent of cars come from agency 3.
It is also estimated that [$pt1perc] percent of cars from agency 1 need a tune-up,
[$pt2perc] percent of cars from agency 2 need a tune-up, and [$pt3perc] percent
of cars from agency 3 need a tune-up. Answer the following questions,
rounding your answers to two decimal places where appropriate.

a) What is the probability that a rental car delivered to the firm will need a tune-up?
[_______]{$q1ans->cmp(tol=>0.01)}

b) If a rental car delivered to the firm needs a tune-up, what is the
probability that it came from agency 2? [_______]{$q2ans->cmp(tol=>0.01)}
END_PGML

#########################################################
## Solution (Optional but recommended)
Context()->texStrings;
BEGIN_SOLUTION
We have the following probabilities, where P (A|B) means the probability of event A given event B:
$BR
$BR
$BCENTER
\(
P(\text{Agency}~ 1) = $p1[0]
\)
$ECENTER
$BR
$BCENTER
\(
P(\text{Agency}~ 2) = $p2[0]
\)
$ECENTER
$BR
$BCENTER
\(
P(\text{Agency}~ 3) = $p3[0]
\)
$ECENTER
$BR
$BCENTER
\(
P( \text{Tune-up required} |\text{Agency} ~1) = $pt1[0]
\)
$ECENTER
$BR
$BCENTER
\(
P( \text{Tune-up required} |\text{Agency}~ 2) = $pt2[0]
\)
$ECENTER
$BR
$BCENTER
\(
P( \text{Tune-up required} |\text{Agency}~ 3) = $pt3[0]
\)
$ECENTER
$BR
$BR
$BBOLD (a) $EBOLD We require P (Tune up required), which is found by adding the
probabilities of the events {Tune up required and car from agency i} over \(i = 1, 2, 3\). For instance,
$BR
$BR
$BCENTER
\begin{align*}
P ( \text{Tune up required and car from agency} 1) &=P(\text{Tune up required} ~ | \text{car from agency} 1) \times P(\text{car from agency} 1) \\
&= $p1[0] \times $pt1[0] =$p11[0].
\end{align*}
$ECENTER
$BR
Finding the probabilities for the other two agencies, we have
$BR
$BR
$BCENTER
\begin{align*}
P(\text{Tune up required})
&= \sum\limits_{i=1}^3 P ( \text{Tune up required and car from agency}~ i) \\
&= \sum\limits_{i=1}^3 P(\text{Tune up required} ~ | \text{car from agency}~ i) \times P(\text{car from agency}~ i) \\
&= $p1[0] \times $pt1[0] + $p2[0] \times $pt2[0] + $p3[0] \times $pt3[0] \\
&= $q1ans.
\end{align*}
$ECENTER
$BR
$BR

$BBOLD (b) $EBOLD Using Bayes Theorem, P (Car from agency 2 | Car requires tune up) is
$BR
$BR
$BCENTER
\(
\frac{ P(\text{Car from agency}~ 2) P(\text{Tune up required}~|~\text{car from agency}~ 2) }{ P(\text{Tune up required}) } = \frac{$p2[0] \times $pt2[0]} { $p1[0] \times $pt1[0] + $p2[0] \times $pt2[0] + $p3[0] \times $pt3[0] } =$q2ans.
\)
$ECENTER
END_SOLUTION
BEGIN_PGML_SOLUTION
We have the following probabilities, where [`\operatorname{P}(A\mid B)`] means the probability of event A given event B:

* [``\operatorname{P}(\text{Agency}~ 1) = [$p1[0]]``]
* [``\operatorname{P}(\text{Agency}~ 2) = [$p2[0]]``]
* [``\operatorname{P}(\text{Agency}~ 3) = [$p3[0]]``]

* [``\operatorname{P}(\text{Tune-up required} \mid \text{Agency} ~1) = [$pt1[0]]``]
* [``\operatorname{P}(\text{Tune-up required} \mid \text{Agency} ~2) = [$pt2[0]]``]
* [``\operatorname{P}(\text{Tune-up required} \mid \text{Agency} ~3) = [$pt3[0]]``]

a) We require [`\operatorname{P}(\text{Tune-up required})`], which is found by adding the
probabilities of the events {Tune up required _and_ car from agency [`i`]} over [`i = 1, 2, 3`]. For instance,

[```
\begin{align*}
\operatorname{P}(\text{Tune up required and car from agency} ~1) &= \operatorname{P}(\text{Tune up required} \mid \text{car from agency} ~1)
\times \operatorname{P}(\text{car from agency} ~1) \\
&= [$p1[0]] \times [$pt1[0]] = [$p11[0]].
\end{align*}
```]

Finding the probabilities for the other two agencies, we have

[```
\begin{align*}
\operatorname{P}(\text{Tune up required})
&= \sum\limits_{i=1}^3 \operatorname{P}( \text{Tune up required and car from agency}~ i) \\
&= \sum\limits_{i=1}^3 \operatorname{P}(\text{Tune up required} \mid \text{car from agency}~ i)
\times \operatorname{P}(\text{car from agency}~ i) \\
&= [$p1[0]] \times [$pt1[0]] + [$p2[0]] \times [$pt2[0]] + [$p3[0]] \times [$pt3[0]] \\
&= [$q1ans].
\end{align*}
```]

b) Using Bayes Theorem, [`\operatorname{P}(\text{Car from agency 2} \mid \text{Car requires tune up})`] is
[```
\frac{ \operatorname{P}(\text{Car from agency}~ 2) \operatorname{P}(\text{Tune up required}\mid \text{car from agency}~ 2)}
{\operatorname{P}(\text{Tune up required})} = \frac{[$p2[0]] \times [$pt2[0]]}{[$p1[0]] \times [$pt1[0]] + [$p2[0]] \times [$pt2[0]] + [$p3[0]] \times [$pt3[0]] } = [$q2ans].
```]
END_PGML_SOLUTION
ENDDOCUMENT();

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