** Using WeBWorK **

# DESCRIPTION  11

# Probability

# WeBWorK problem written by Tim payer, <tsp1@humboldt.edu>

# ENDDESCRIPTION

## DBsubject(Probability)

## DBchapter(??)

## DBsection(??)

## Institution(Humboldt State University)

## Author(Tim Payer)

## KEYWORDS(probability, addition, notation)

DOCUMENT();

loadMacros(

"PGstandard.pl",

"MathObjects.pl",

"PGML.pl",

"parserFormulaUpToConstant.pl",

"niceTables.pl",

"PGunion.pl",

"parserPopUp.pl",

);

Context("Numeric");

Context()->flags->set(

  tolerance => 0.0001,

  tolType => "absolute",

);

# \{row("Income $", @z)\}

# Table data generation:

# To switch off the algorithmic table data un-comment the following 5 lines:

# and then Comment (#) the random assignments.

# $r[4] = 12;

# $u[4] = 150;

# $m[4] = 200;

# $l[1] = 200;

# $p[1] = 50;

# 

$r[4] = random(12, 18,1);

$r[3] = $r[4] + 50;

$r[2] = $r[4]*10 + 6;

$r[1] = $r[4]*30 - 7;

$r[0] = $r[4]*55 + 22;

$rsum = $r[4]+$r[3]+$r[2]+$r[1]+$r[0];

$u[4] = random(121, 161,1);

$u[3] = $u[4] + 121;

$u[2] = $u[4]*4 + 6;

$u[1] = $u[4]*17 - 7;

$u[0] = $u[4]*5 + 22;

$usum = $u[4]+$u[3]+$u[2]+$u[1]+$u[0];

$m[4] = random(191, 211,1);

$m[3] = $m[4]*4 + 22;

$m[2] = $m[4]*10 + 17;

$m[1] = $m[4]*6 - 24;

$m[0] = $m[4]*2 + 51;

$msum = $m[4]+$m[3]+$m[2]+$m[1]+$m[0];

$l[0] = random(181, 213,2);

$l[1] = $l[0]*2 + 89;

$l[2] = $l[0]*3 + 17;

$l[3] = $l[0]*10 + 24;

$l[4] = $l[0]*3 + 51;

$lsum = $l[4]+$l[3]+$l[2]+$l[1]+$l[0];

$p[0] = random(41, 58,1);

$p[1] = $p[0] + 22;

$p[2] = $p[0]*2 + 17;

$p[3] = $p[0]*3 - 15;

$p[4] = $p[0]*6 + 51;

$psum = $p[4]+$p[3]+$p[2]+$p[1]+$p[0];

# Column sums:

$c0 = $p[0]+$l[0]+$m[0]+$u[0]+$r[0];

$c1 = $p[1]+$l[1]+$m[1]+$u[1]+$r[1];

$c2 = $p[2]+$l[2]+$m[2]+$u[2]+$r[2];

$c3 = $p[3]+$l[3]+$m[3]+$u[3]+$r[3];

$c4 = $p[4]+$l[4]+$m[4]+$u[4]+$r[4];

$grand = $c0+$c1+$c2+$c3+$c4 +$psum+$lsum+$msum+$usum+$rsum;

$popup1 = PopUp(

["probability notation", "P(86-88)", "P(A)", "P(B)", "P(C)", "P(D)","P(E)"], "P(B)");

$ans1 =$c1/$grand;

$popup2 = PopUp(

["probability notation", "P(B U F)", "P(BF)", "P(B ~~x{2229} F)", "P(B + F)", "P(B x F)"], "P(B ~~x{2229} F)");

$ans2 =$p[1]/$grand;

# Apparently the unicode wont work here?  "~~x{2229}" wont render

BEGIN_TEXT 

$PAR

4.2)  Recent census data has correlated life expectancy with one's income bracket. $BR

Given a small town of \($grand\) individuals, that is loosely based upon the US census data, determine the following probabilities using correct probability notation. Answer with fourth decimal accuracy and use the event variables declared below:

$PAR

Where:  $BR

$PAR

$BLEFT

\{ begintable(7) \}

\{ row( "Life Expectancy", "89-95", "86-88", "81-85","77-80","74-76" ,"Row Sums") \}

\{ row( "Rich:", $r[0], $r[1],$r[2], $r[3], $r[4], $rsum ) \}

\{ row( "Upper Middle Class", $u[0], $u[1],$u[2], $u[3], $u[4], $usum) \}

\{ row( "Middle Class", $m[0], $m[1],$m[2], $m[3], $m[4], $msum) \}

\{ row( "Lower Middle Class", $l[0], $l[1],$l[2], $l[3], $l[4], $lsum) \}

\{ row( "Poor", $p[0], $p[1],$p[2], $p[3], $p[4], $psum) \}

\{ row( "Column Sums:", $c0, $c1,$c2, $c3, $c4, $grand) \}

\{ endtable() \}

$ELEFT

$PAR

\(A \) = Event that an individual has a life expectancy of 89 - 95 years. $BR

\(B \) = Event that an individual has a life expectancy of 86 - 88 years. $BR

\(C \) = Event that an individual has a life expectancy of 81 - 85 years. $BR

\(D \) = Event that an individual has a life expectancy of 77 - 80 years. $BR

\(E \) = Event that an individual has a life expectancy of 75 - 76 years. $BR

\(R \) = Event that an individual is rich. $BR

\(S \) = Event that an individual of the upper middle class $BR

\(M \) = Event that an individual is of the middle class. $BR

\(L \) = Event that an individual is of the lower middle class.$BR

\(F \) = Event that an individual is poor. $BR

END_TEXT

BEGIN_PGML

4.2a)  Find the probability of drawing an individual from the town that has a life expectancy of 86-88 years:  

[_____]{$popup1} = [______]{$ans1}

4.2b)  Find the probability of drawing an individual from the town that has a life expectancy of 86-88 years and is poor:  

[_____]{$popup2} = [______]{$ans2}

END_PGML

BEGIN_PGML_SOLUTION

*SOLUTION*

To simplify the expression to a single term with a rational denominator we first start by splitting the root between the numerator and denominator. Then find a common denominator and rationalize the denominator. 

 

[``\begin{aligned}&\\  

[$c]-\sqrt{\frac{[$am] x}{[$bm]}} &=[$c] -\sqrt{\frac{[$m]\cdot[$a] x}{[$m]\cdot[$b]}} = [$c]  -\frac{\sqrt{[$a] x}}{\sqrt{[$b]}}

&& \text{Cancel the common factor and then split the root.}\\    

&=\frac{[$c]\sqrt{[$b]}}{ \sqrt{[$b]}}-\frac{\sqrt{[$a] x}}{\sqrt{[$b]}} && \text{Create a common denominator.}\\  

&=\frac{[$c]\sqrt{[$b]} -\sqrt{[$a] x}}{\sqrt{[$b]}} && \text{Combine in to a single fraction.}\\    

&=\frac{\left([$c]\sqrt{[$b]} -\sqrt{[$a] x}\right)\sqrt{[$b]}}{\sqrt{[$b]}\cdot\sqrt{[$b]}} && \text{Multiply the numerator and denominator by the denominator.}\\  

&=\frac{\left([$c]\sqrt{[$b]\cdot [$b]}-\sqrt{[$a] \cdot [$b] x}\right)}{[$b]} && \text{Distribute through the numerator and reduce the denominator.}\\  

&=\frac{\left([$c] \cdot [$b] -\sqrt{[$ab] x}\right)}{[$b]} && \text{Reduce the square root factors.}\\  

&=\frac{[$cb] - \sqrt{[$ab] x}}{[$b]} && \text{Combine numerator factors.}  

\end{aligned}``]  

END_PGML_SOLUTION

ENDDOCUMENT();