\chapter{Ratios, Percents, Proportions}

\specialsection{Introduction}
Measurements sometimes have meaning in isolation.  For instance,  
on June 21, 2017 as I was writing this in Worcester, MA, the day was sunny and the high temperature 
was $81^\circ$ F.  You can see from that information that it was a very comfortable early summer
day(!)  However, there are also times when we want to \emph{compare} measurements or other
quantities in various ways.  Here is an everyday example that will illustrate some of
the key points we want to make in this chapter.\footnote{Depending on how comfortable the class is with this material, 
we might omit covering some or all of this in class.}  

According to a news story published in the {\it New York Daily News} on January 4, 2017\footnote{
Archived at {\tt http://www.nydailynews.com/new-york/nyc-crime/ \\
nyc-historically-number-shootings-2016-article-1.2933098}, 
consulted June 21, 2017.}
there were $352$ homicides reported in New York City during the calendar year 2015 versus $335$ homicides during the 
calendar year 2016.  
In looking at the homicides from one year to the next, the most basic comparison 
would be to say $335 - 352 = -17$, so the number of homicides \emph{decreased by $17$}.  But we could also  
ask more about this situation: Is that a big decrease or a small decrease?  One way to answer that
more refined question would be to compute what is called the \emph{percent decrease}.  The idea here is that 
we look at the decrease \emph{as a fraction of the number of homicides in the previous year}, then express 
that fraction as a \emph{percent} by giving it as a number out of the standard nominal total of 100\%.\footnote{This is 
the origin of the word ``percent'' since 100 is ``centum'' in Latin.  The $\%$ sign was invented for use in accounting
and in specification of interest rates.}
The computation would be 
\begin{equation}
\label{eq:HomicidePercentDec}
{\rm percent\ decrease} = \frac{335 - 352}{352} \times 100\% \doteq -4.8\%.
\end{equation}
A decrease of $4.8\%$ would usually be interpreted as a ``medium size'' decrease--not huge, but 
not negligibly small either.   The homicide figures were moving in the right
direction in 2016, but the change was incremental rather than massive.

On the other hand,\footnote{According to {\tt https://www.neighborhoodscout.com/ar/crime}, 
consulted June 21, 2017.}  there were 181 homicides in the state of Arkansas in calendar year 2016
versus the 335 homicides reported in New York City that year.  
Looking at the New York City homicide figure versus the Arkansas state
homicide figure, it might be tempting to think that New York City was measurably less safe
and more crime-ridden than Arkansas.  After all, $335$ is only a bit less than twice the Arkansas
figure since  $2 \times 181 = 362$, 
and the $181$ is the figure for a whole state, not just one city!  

But we have to ask: Is this is a reasonable way to make the comparison?  Or might we be
skewing the data in some way by just comparing the numbers of homicides?  Before you read on, 
ask yourself: Might we be leaving out some relevant information?

In 2016, the population of Arkansas was approximately $2.98$ million people, while the population 
of New York City was approximately $8.54$ million people.  If we take that difference into account 
by computing the \emph{per capita homicide rates}, that is, 
the number of homicides per person\footnote{``Capita'' comes from the Latin word for ``head.''}
in the populations, we get approximately the following figures:
$${\rm New\ York\ per\ capita\ homicide\ rate} = \frac{335}{8.54 \times 10^6} \doteq .000039,$$
and
$${\rm Arkansas\ per\ capita\ homicide\ rate} = \frac{181}{2.98\times 10^6} \doteq .000063.$$
These are both quite small numbers, of course, since the number of homicides was a very small
fraction of the total population in either case.  However, there is also a comparison we could
make between the two per capita rates by computing a \emph{percentage difference}--that is
by expressing the difference between the per capita rates in Arkansas and New York City 
as a fraction of the per capita rate in New York City, then converting that to a percentage.   
By that measure,  the rate in Arkansas was actually
\begin{equation}
\label{eq:PercentageDifference}
\frac{.000063 - .000039}{.000039} \times 100\% = 61.5\%
\end{equation}
higher than the rate in New York City.  This percentage difference
would be usually be interpreted as being relatively high.  
By this measure, New York City was actually quite a bit less dangerous than
Arkansas!\footnote{I'm not trying to pick on Arkansas here.  There are 
a number of other relatively small states where the comparison would work
out similarly!}

Another possible way to interpret these numbers is that in New York City, if you picked a 
person at random from a list of the population, then there would be a $.0039\%$ chance that person was
murdered in 2016.  Similarly, there was $.0063\%$ chance a randomly chosen Arkansan
was murdered.  One possible objection to this way of thinking about the numbers, though,
is that we don't know whether all of the homicide victims were natives or whether there might
have been visitors included in those homicide totals.  So
this way of thinking about the numbers is slightly suspect in this particular case.  Also, to make a fairer 
comparison between crime levels in the two locations, we would probably also want to look at rates of other 
violent and non-violent crimes, not just homicides.  But we could use the same methods to make those
comparisons too and probabilities are often computed by the same sort of ratios that we have been 
discussing in this example.  In this chapter, we will review the ideas of ratios, proportions,
and percents that we have been using here and see many of their applications. 

\specialsection{Ratios and Percentages}
From the mathematical point of view, taking a ratio always involves \emph{dividing} one
numerical measurement by another numerical measurement.  The units of the two measurements
can be the same or different, depending on the situation.  

If the units of the two quantities are the same, then the units in the numerator and 
the denominator of the ratio cancel, and the value of the quotient gives what
is called a \emph{dimensionless quantity}, or pure number.  The meaning of the
ratio in this case is usually just a simple comparison of the sizes of the numerator
and denominator in the quotient.  

\begin{example}
The formula for computing sound pressure levels
$$L = 20 \times \log_{10}\left(\frac{p_m}{p_r}\right).$$
from Exercise~\ref{ex:SoundPressure} in Chapter 1 involves
the quotient $\frac{p_m}{p_r}$ where the reference pressure $p_m$
and the measured pressure $p_m$ have the same units (micropascals).
Hence the quantity $L$ will also be a pure number, or dimensionless
quantity. $\triangle$
\end{example}

\emph{Percentages} are also frequently used to represent one quantity 
as a fraction or multiple of some other reference quantity, measured with the same units.  
Any ratio where the numerator and denominator have the same units can be converted into an 
equivalent percentage of the denominator value simply by 
multiplying the numerical value of the quotient by $100\%$.  The effect is to convert the 
ratio into another equivalent ratio with an (implicit) denominator of $100$.  
Thus if a first quantity is $25\%$ of a second quantity, we see that the first is 
$$\frac{25}{100} = \frac{1}{4}$$ 
of the second.  Similarly, if the first quantity is $150\%$ of the second, the 
first is 
$$\frac{150}{100} = \frac{3}{2}$$
of the second, or ``half again as large'' as the second.  

\begin{example}
\label{Homicides}
For example, in the data presented in the Introduction to this chapter, if we
computed the ratio of the number of homicides in New York to the number of 
homicides in Arkansas (both in 2016), we would obtain
$$\frac{335}{181} \doteq 1.85.$$
The $1.85$ is a pure number which tells us that the raw number
of homicides in New York was $1.85$ times as large, or almost twice as large, as the number of 
homicides in Arkansas.  We could also say that the number of homicides in New York 
City in 2016 was 
$$\frac{335}{181} \times 100\% \doteq 185\%$$
of the number of homicides in Arkansas that year.  $\triangle$
\end{example}  

Percentages also feature in the computation of \emph{percentage changes} (sometimes
phrased as \emph{percentage differences}).  
These ideas are notoriously ``slippery''\footnote{That is, easy to get wrong and/or
misunderstand.} for many people, 
so please make sure that you understand them in detail, as we will be using them
repeatedly throughout the course.  The situation is always something like this: 
we measure some quantity twice,  yielding an ``old value'' and an updated, ``new value,''
or we consider an operation on the old value that changes it into the new value.  
The percentage change is a way to represent the difference between the two values.  It is 
\emph{the change as a percentage of the ``old value.''} To compute the 
percentage change, we always use
\begin{equation}
\label{eq:PercentChange}
\framebox{$\displaystyle{{\rm percentage\ change} = \frac{{\rm new\ value} - {\rm old\ value}}{\rm old\ value} \times 100\%}$}
\end{equation} 
If the new value is larger than the old value, the quotient will be positive and the percent change
will represent a percent increase; if the new value is smaller, we will have a percent decrease. 
I believe that the ``slipperiness'' mentioned before comes mainly from the fact that we need to identify
which quantity represents the old value and which represents the new value when
we apply the formula.  This requires some ``common sense'' in many cases -- you can't just
fly blind; you need to think through which is which.  

For instance, in \eqref{eq:PercentageDifference}, 
note that we treated the New York City per capita rate as the ``old value'' because we
wanted to think of the Arkansas per capita rate as the ``new value'' and compare that to the 
New York City rate.  It would be possible to make the comparison the other way too, but
then the description would change and the formula would be used with the values reversed.  
The percentage difference
$$\frac{.000039 - .000063}{.000063} \times 100\% \doteq -38.1\%$$ 
would say that the New York City per capita homicide rate is about $38.1\%$ smaller than the Arkansas
rate.  In the context of \emph{percentage differences}, it may in fact help to think 
of the ``old value'' as a \emph{reference value} and the ``new value'' as a \emph{comparison
value} that is being compared to the reference value.   

We start with some ``everyday'' examples.

\begin{example}
If a man's suit that regularly sells for $\$ 300$ (the old value) is offered 
on sale for $\$ 200$ (the new value), this is a 
$$\frac{200 - 300}{300} \times 100\% \doteq -33.3\%$$
percentage decrease or discount.  On the other hand, if we knew the suit
was being sold at a $25\%$ discount, this should mean that the reduced price (the new value) 
is some number $\$ x$  where
$$\frac{x - 300}{300} \times 100\% = -25\%.$$
If you do the algebra to solve for $x$ (or even better, think: what is $25\%$ of $\$ 300$? then
subtract that from $300$), you will see that the reduced price must be $x = \$ 225$.  This is different from 
taking $x$ plus $25\%$ of $x$.  We aren't taking $25\%$ of the \emph{old value (price)} that way! 

Similarly, suppose the wholesale price of a refrigerator is $\$ 500$, but the appliance store sells it with
a $20\%$ markup.   Now the old value is the wholesale price, and the retail price, which equals the 
wholesale price plus the markup, is the new value.  The retail price must then be $\$ 600$, since 
$$\frac{600 - 500}{500} \times 100\% = 20\%$$
gives the $20\%$ increase. $\triangle$
\end{example}

We saw another similar example in the Introduction too.

\begin{example}
Look back at Equation~\eqref{eq:HomicidePercentDec} in the Introduction to this chapter.
There the new value was the 2016 homicide total of $335$ and the old value was the 2015 homicide total
of $352$ and the decrease of $17$ homicides represented a percentage decrease
of approximately $4.8\%$.  $\triangle$
\end{example}

Here is an example related to the Chapter Project from Chapter 1.

\begin{example}
\label{GreenlandIceMaxDepth}
The maximum depth of the Greenland ice sheet shown in Figure~\ref{fig:Greenland} from Chapter 1 is
about $3205$ meters.  This comes from depth estimates made about 2013.  Earlier estimates
reported by Bamber, Layberry, and Gogineni\footnote{Journal of Geophysical Research, vol. 106, D24, December 27, 2001,
pp. 33,773-33,780.} gave the maximum depth as $3367$ meters.  This estimate was based on observations
from the 1970's through the 1990's, so it will serve as our ``old value.''  We will take the ``new value'' 
to be the $3205$ meters.  This gives a percent change of
$$\frac{3205 - 3367}{3367} \times 100\% \doteq -4.8 \%,$$
(coincidentally) another percentage decrease of about $4.8\%$.  

Some care should be exercised in interpreting this result.
This might represent an actual decrease in the thickness of the Greenland ice sheet.  
But it might also conceivably simply reflect more accurate measurements yielding
smaller depth estimates.  We have not presented enough information to make a firm
determination of which of these alternatives is more probably correct.
 However, on the face of it, it seems likely that the thickness has actually decreased, given
the fact that large melting events have been observed from the early 2000's to the present.  $\triangle$  
\end{example}

A closely related idea is the \emph{percentage relative error} in a measurement
when an exact theoretical value is available:
\begin{equation}
\label{eq:PercentRelativeError}
\framebox{$\displaystyle{{\rm percentage\ relative\ error} = \frac{{\rm measured\ value} - {\rm exact\ value}}{\rm exact\ value} \times 100\%}.$}
\end{equation} 
This is arranged to give positive results when the measured value is larger than the exact value
and negative results when it is less.  

\begin{example}
\label{RelError}
Suppose an object's mass is exactly $34.7$ kg, but we measure its mass to be $35$ kg.  The 
percentage relative error in this measurement is 
$$\frac{35 - 34.7}{34.7} \times 100\% \doteq .86\%.$$
The absolute error (the actual amount by which the measurement differs from the exact
value) in this case is $.3\ {\rm kg}$ which is not an especially
small mass.  Nevertheless, we would say this was a rather \emph{accurate}
measurement in relative terms because the error is a very small fraction 
(less than $1\%$) of the exact value. $\triangle$
\end{example}

So far we have been discussing ratios where the numerator and denominator
have the same units.  But ratios where the numerator and denominator have different
associated units are also very common and useful.  In this case the 
quotient value also comes with associated units that are 
something new and different from either the units of the numerator 
or the denominator.  If the units of the numerator are $N$ and the 
units of the denominator are $D$, then we call the units of the quotient
``$N$ per $D$,'' and write this as $N/D$.  For instance, if the numerator was
the total cost of a hotel stay in units of $\$$ and 
the denominator was a length of time in days, then
the ratio would have units of $\$$/day, or dollars per day.

\begin{example}
\label{PopDensity}
As we said before, the population of New York City in 2016 was about 
$8.54 \times 10^6$ people.  The city occupies a land area of about 
$302.6$ square miles.  The ratio
$$\frac{\rm population}{\rm land\ area} \doteq \frac{8.54 \times 10^6}{302.6} \doteq 
28,222\ {\rm people/square\ mile}$$
is called the (average) \emph{population density} of the city.  The 
meaning is this:  If the total population was spread evenly over the 
whole area\footnote{This is not actually the case.  Manhattan 
is more densely populated than the four other New York City boroughs.  
See Exercise~\ref{ex:NYBoroughPDs} below.} 
then there would be over $28,000$ people per square
mile.  This is an extraordinarily large population density, of course, showing
the way multi-story apartment buildings and all the associated infrastructure  
of a large modern city can allow large numbers of people to live within a small 
geographic area.  
$\triangle$
\end{example}

If you have studied velocities in a physics or calculus class, the following
example will be familiar.  

\begin{example}
\label{AveVel}
If a moving object travels along a straight line path for a distance of $\Delta x = 34$ m
in a time $\Delta t = 5$ sec, then we say it is moving at the (average) \emph{velocity}
$$v_{ave} = \frac{\Delta x}{\Delta t} = \frac{34}{5} = 6.8\ {\rm m/sec}.$$
The ratio $\frac{\Delta x}{\Delta t}$ computes \emph{the distance traveled per second}, 
if the object moves at the same speed for the whole time (or the average distance traveled
per second if not).  Note that the units here are m/sec (``meters per second''). $\triangle$
\end{example}

One purpose of ratios like the ones giving the average population density of 
cities is that they make it possible to compare different examples of cities on 
an ``apples to apples'' basis, independent of the raw total population figures.  
The population density can have a greater influence on the quality of life and ``feel''  
of a city than the raw total population figure.   For example, you will see in 
Exercise~\ref{ex:CityPDs} that even though Chicago has a much larger total population 
than Boston, that population is not as densely packed because the total area of Chicago
is also much, much larger.  

\specialsection{Concentrations}
Environmental scientists often use ratios together with a variation on the 
notion of percents to describe the \emph{concentrations} of pollutants or contaminants in air, 
water, foods, etc.  The idea is described in the following example.

\begin{example}
\label{Fish}
Burning of coal and some other industrial processes release small amounts of mercury and other
heavy metals into the atmosphere.  That mercury is then incorporated into the bodies 
of fish and land animals as they eat plants, smaller fish, and smaller animals that have been
contaminated with it.  
As a result, animals at the tops of food chains tend to concentrate mercury in their 
tissues.  Mercury levels in fish are of concern for human health because fish is
an increasingly important part of many human diets and ingesting
even very small amounts of the metal can damage the central and peripheral nervous system.  

According to the U.S. Food and Drug Administration\footnote{
{\tt https://www.fda.gov/food/foodborneillnesscontaminants/metals/ucm115644.htm}, consulted 
June 23, 2017.}, the flesh of the following popular shellfish and fish caught for 
human consumption as food (data collected through 2012) contained mercury at
the levels shown in Table~\ref{MercuryInFish}.
\begin{table}[ht]
\caption{Mercury Concentrations in Fish (in parts per million, ppm)}
\label{MercuryInFish}
\renewcommand\arraystretch{1.5}
\noindent\[
\begin{array}{|c|c|}
\hline
{\rm Species} & {\rm Concentration\ in\ ppm} \\
\hline
{\rm Scallop}& .003  \\
\hline
{\rm Shrimp}&  .009 \\
\hline
{\rm Salmon\ (canned)}& .014 \\
\hline
{\rm Salmon\ (fresh,\ frozen)} & .022 \\
\hline
{\rm Catfish }& .024 \\
\hline
{\rm Lobster} & .107 \\
\hline
{\rm Tuna\ (canned,\ light) }& .126 \\
\hline
{\rm Tuna\ (canned,\ white) }& .350 \\
\hline
{\rm Grouper} & .448 \\
\hline
{\rm Swordfish} & .995 \\
\hline
\end{array}
\]
\end{table}
The units used here (parts per million by mass) represent the contaminant (the mercury)
as a fraction of the total mass of the fish.  This gives a common way to represent
very small amounts without using extremely small mass or volume values.  Those very small
values might cause confusion and misunderstanding.  

What this means, for instance, is that flesh from a type of fish would contain 
1 part per million 
mercury by mass if a $1,000,000$ g sample of the fish contained $1$ g of mercury:
$$\frac{1\ {\rm gram\ mercury}}{1,000,000\ {\rm grams\ fish}} = 1\ {\rm ppm}.$$
The swordfish mercury level in the table above is nearly that high; the
grouper mercury level as about one half that high; and the catfish
mercury level is about $1/40$ times that high.   

The values in Table~\ref{MercuryInFish} might seem extremely small in 
everyday terms and we might be tempted to think that there is nothing
to worry about.  The bad news is that it takes very little mercury
to produce the nervous system damage we mentioned before.  In fact,
on the basis of this data, the Environmental Protection Agency
recommends that children (who are more susceptible to mercury poisoning
because of their smaller body weights and developing nervous tissue)
eat white (albacore) tuna no more than \emph{once a month} based on 
this level of mercury!  Adults are recommended to limit their intake 
of white (albacore) tuna to about \emph{three times a month}. $\triangle$
\end{example}
  
Larger concentrations could
also be reported using units of parts per thousand; smaller amounts might
be reported using units of parts per billion.  The idea is the same.

\specialsection{Probabilities}
One of the commonly-used conceptual frameworks for thinking about probabilities,
called the \emph{frequentist approach}, uses ratios to define and compute
probabilities.\footnote{There is also an alternative school of thought in 
probability and statistics known 
as the \emph{Bayesian} approach, where probabilities are treated as estimates
and updated on the 
basis of the information supplied by experiments. This is increasingly
important in scientific applications and we may be able to discuss some of the
basics of Bayesian probability in the spring semester.}

For the purposes of this introductory discussion, we will stick to situations
where the total number of possible outcomes of a measurement or an experiment
is a \emph{finite set} called the \emph{sample space}, $S$.  
It is often the case that each single element of the sample space is
considered to be equally likely as an outcome.  If so, then we assign a 
\emph{probability} of 
$$\frac{1}{{\rm number\ of\ elements\ of}\ S}$$
to each element of the sample space.  Note that each of these values
is $\ge 0$ and adding up all of them (i.e for each element of $S$) gives
a total of $1$.  If the elements of the sample space are not all equally
likely as outcomes, then each would be assigned some probability $\ge 0$, 
and these would again sum to a total value of $1$.   Either way, we
have what is called a \emph{probability space}, or \emph{probability model}
consisting of the sample space and the assignment of probabilties.  

\begin{example}
\label{FishPool}
Suppose that our experiment consists of selecting one randomly 
chosen fish out of a pool containing $13$ carp, $10$ sunfish
and $3$ northern pike.  If each fish is equally likely as the one selected, 
then the probability of picking any one individual fish
would be $\displaystyle{\frac{1}{13 + 10 + 3} = \frac{1}{26}}$.    $\triangle$
\end{example}

An \emph{event} is a subset $E$
of the sample space.  Using the assignment of probabilities to the elements of $S$,
we define the probability of $E$, written
$P(E)$, as \emph{the sum of the probabilities of the elements of $E$}\footnote{Technical note:
Recall that we are restricting to the case where $S$ is a finite set.  If $S$ is an interval
on the real line or a region in the plane or in a higher-dimensional space, then probabilities
are often described by a corresponding \emph{probability density function} and techniques
from integral calculus can be used to compute them.  Many Calculus 2 courses
study those applications of integration to computing probabilities.}   In the
equal-probability case, this gives
\begin{equation}
\label{eq:ProbDef}
P(E) = \frac{\rm number\ of\ elements\ of\ E}{\rm number\ of\ elements\ of\ S}
= \frac{{\rm outcomes\ in\ } E}{{\rm outcomes\ in\ } S}.
\end{equation}
This should be come more understandable if we pause for some examples to 
illustrate these ideas. 

\begin{example}
\label{FishPoolContinued}
Continuing from Example~\ref{FishPool}, suppose that $E$ represents
the outcome that we select a sunfish.  Since each of the fish is
equally likely as the one selected and there are 10 of the sunfish, the probability of 
$E$ would be obtained by adding together $10$ terms equal to $\frac{1}{26}$:
$$P(E) = \frac{1}{26} + \cdots + \frac{1}{26} = \frac{10}{26} = \frac{5}{13} \doteq .385. \quad
\triangle$$
\end{example} 

Here is another, somewhat more involved, example.

\begin{example}
\label{TwoDice}
Suppose you roll two cubical (six-sided) dice, one with black dots on
a white background, the other with white dots on a black background.
Each of the dice will come up with one face facing upward, showing
a number of dots between $1$ and $6$.  Because the two dice are
visually different, note that we can choose to record the results 
by showing the number from the die with black dots first, then 
the number from the die with white dots second.  This means that 
in mathematical terms, 
the possible outcomes of one roll are in one-to-one correspondence
with the \emph{ordered pairs}
$(m,n)$ where $m$ is $1,2,3,4,5$, or $6$ and similarly
for $n$.  This is the sample space $S$ for the experiment of 
rolling the two dice once, and we could list all of its $6 \times 6 = 36$
elements like this:
$$S = \{(1,1),(1,2),\ldots,(1,6),(2,1),(2,2),\ldots,(2,6),\ldots,(6,1),(6,2),\ldots,(6,6)\}.$$

In this simple example, if the two dice are both ``fair,'' 
that is, if they are perfectly balanced and not ``loaded'' so that some 
numbers come up more frequently than others, then it would make
sense to assign the same probability $\frac{1}{36}$ to each of the 
elements of $S$ (the possible outcomes of the roll).  

On the other hand, 
suppose the two dice were both loaded in the same way so that $3$'s came up 
$\frac{1}{5}$ of the time but each of the other numbers
came up $\frac{4}{25}$ of the time.  Note that 
$$\frac{1}{5} + 5\times \frac{4}{25} = \frac{1}{5} + \frac{4}{5} = 1,$$
as we required above.  Then the probability of rolling
a $3$ on both dice is
\begin{equation}
\label{eq:Loaded1}
P((3,3)) = \frac{1}{5}\times \frac{1}{5} = \frac{1}{25} = 0.04.
\end{equation}
The probability of rolling one 3 and something else (that is, $(3,n)$
with $n\ne 3$ or $(m,3)$ with $m \ne 3$) is: 
\begin{equation}
\label{eq:Loaded2}
P((3,n)) = P((m,3)) = \frac{1}{5}\times \frac{4}{25} = \frac{4}{125} = 0.032.
\end{equation}
Finally, if $m,n$ are both different from $3$, 
\begin{equation}
\label{eq:Loaded3}
P((m,n)) = \frac{4}{25} \times \frac{4}{25} = \frac{16}{625} = .0256.
\end{equation}
We see that rolling two threes is more likely than either of the other two
possibilities.ß

Now, let us consider the event $E$ described in words by saying
that the die with black dots shows an \emph{even number}, while
the die with white dots shows an \emph{odd number}.  As a subset of $S$,  
$$E = \{(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),(6,1),(6,3),(6,5)\}.$$
Hence if the dice are not loaded, we will have
$$P(E) = \frac{\rm number\ of\ elements\ of\ E}{\rm number\ of\ elements\ of\ S} = \frac{9}{36} = \frac{1}{4}.$$
In the exercises, you will compute $P(E)$ if the dice are
``loaded'' as above.  $\triangle$
\end{example}

The frequentist interpretation of the \emph{meaning of these calculations}
is that if we repeated the experiment for a large number of trials and 
kept track of the ratio
\begin{equation}
\label{eq:FreqProb}
\frac{{\rm number\ of\ outcomes\ in\ } E}{{\rm number\ of\ trials}},
\end{equation}
then this ratio would tend to approach $P(E)$ as the number of trials increased
without bound.  This statement is often called the \emph{Law of Large Numbers}.  
For example, with the two fair dice as described in Example~\ref{TwoDice},
if we roll the dice a large number of times, then roughly 
$\frac{1}{4} \times 100\% = 25\%$ of the rolls should show
an even number of black dots and an odd number of white dots.   If the
rolls are truly random, there could be more or less than $25\%$ of them meeting
this condition.  However, the deviation from the probability model
value of $25\%$ should decrease the more times we roll the dice.  
Moreover, in the absence of a theoretical probability model as
we described above, a frequentist might take a value computed
as in Equation~\eqref{eq:FreqProb} as an ``experimental probability''
or estimate of the unknown theoretical probability.   


\specialsection{Proportions}
Mathematically, we say a \emph{proportion} is the statement that two ratios
are the same:
$$\frac{A}{B} = \frac{C}{D}.$$
Alternatively, we might say $A$ has the same proportion to $B$ as $C$ has to $D$
if this equation is true.  Note that if we know \emph{three} of the terms
in a proportion, then it is always possible to solve for the remaining
term by simple algebraic manipulations.  

\begin{example}
\label{MapScale}
Any time we estimate
a distance from a map and the associated distance scale we are doing something
similar.  If the map says $1.5$ inch = $10$ miles, and we measure a distance
of $2.5$ inches on the map, then the real world distance $x$ between those
two points will satisfy the proportion 
$$\frac{x}{2.5} = \frac{10}{1.5},$$
so $x = 2.5 \times \frac{10}{1.5} = 37.5$ miles.  Similarly, if we were constructing 
a map with the scale $1.5$ inch = $10$ miles and we knew the actual distance
between two locations to be $45.3$ miles, then the separation between
the points on the map representing those locations would be $x$ satisfying the proportion
$$\frac{45.3}{x} = \frac{10}{1.5},$$
so 
$$x = \frac{45.3 \times 1.5}{10} \doteq 6.8\ {\rm inches}.$$ 
The conversions from one system of units to another from Chapter 1 and the 
calculations of distances and areas we were doing in the Chapter Project
were based on proportions. $\triangle$
\end{example}

The idea of proportionality is also used in the following way. We say 
two quantities $A$ and $B$ \emph{are proportional}, or equivalently \emph{$A$ is proportional
to $B$}, if there is some
constant $c$ such that $A = c B$.  The constant $c$ is called 
the \emph{proportionality factor}. This means that if we know the value
of $B$, then the corresponding value of $A$ is obtained by simply multiplying
by the constant $c$.

\begin{example}
\label{CircleAreaRadius}
The area of a circle is proportional to the square of its radius and also to the 
square of its diameter.\footnote{For those
with an interest in the history of mathematics, this is the way areas of circles 
were treated in many ancient Greek texts, for example in Proposition 2 of Book XII of 
Euclid's \emph{Elements}.  Euclid does not mention the proportionality factor
$\pi$ because he did not use numbers as measures of areas.  The value of $\pi$
was later approximated rather closely by Archimedes.}  This true since if $r$ is the radius,
we have the formula $A = \pi r^2$, and $\pi \doteq 3.14159$ is constant.   
Moreover, if $d$ is the diameter then $r = d/2$, so $A = \frac{\pi}{4} d^2$, 
with proportionality factor $\pi/4$.
$\triangle$
\end{example}

\subsection*{Capture-Recapture}
Proportions are used in a clever way by biologists and environmental scientists
when they want to estimate the population of some species of animals in 
a habitat.\footnote{The method works best when the habitat is \emph{closed}
so there is no in- or out-migration of the animals.}  The motivation 
for this method is that it is usually not practical to do an exhaustive
census of the whole population.  It may be too difficult or time-consuming
to capture and count all of the animals present.  However, it \emph{is often
possible} to capture and tag a random sample of the animals in the population,
then release them back into the population and let them ``mix in'' with the
rest.  After enough time is allowed for mixing (but not too much time, so that
not many births and deaths take place), a
new sample is taken and the proportion of tagged animals in the new 
sample is computed.  Assuming that the animals that were tagged from 
the first sample were mixed thoroughly into the population before 
the second sample was taken we would expect the following proportion
to hold (at least approximately):
\begin{equation}
\label{eq:CaptureRecapture}
\framebox{$\frac{\rm size\ of\ first\ sample}{\rm total\ population} = 
\frac{\rm tagged\ animals\ in\ second\ sample}{\rm size\ of\ second\ sample}$.}
\end{equation}
Three of the terms here, namely all except for the total population, are known. 
But this means we can use
the proportion to estimate the total population.  This process is called
the \emph{capture-recapture, or catch-and-release} method.  

\begin{example}
\label{CREx}
Suppose a first sample of 25 muskrats are captured, tagged and released
in a forest.  A week later, a new sample of 40 muskrats is taken, and
6 of them are found to be animals tagged in the first sample.  
We want to use this information to estimate the population of muskrats
in the forest.  Using Equation~\eqref{eq:CaptureRecapture}, and writing $x$
for the total population, which is the unknown quantity, we see
$$\frac{25}{x} = \frac{6}{40} \Rightarrow x = \frac{25\cdot 40}{6} \doteq 167$$
muskrats.  $\triangle$
\end{example}

\specialsection{Chapter Project}
The project for this chapter has two goals:
\begin{enumerate}
\item To continue our work with ratios, percentages, and proportions on some
world population and GDP data from the U.N., and 
\item To introduce the Excel spreadsheet program\footnote{Some freely available software such as 
the LibreOffice suite can also be used here.} that we will be using extensively
for the rest of the year.  
\end{enumerate}
The following instructions will lead you through a first practice Excel session
working with a spreadsheet file that you will download from the 
course homepage.

\subsection*{Getting Started with Excel}
Launch Excel and 
take a look at the overall layout of the of the window.
There are tabs, menus, etc. similar to many standard
programs, but there are some differences too.  In particular
if you are using the Windows Excel 2007 or later version, 
note the large ``Office Button'' at the upper left.  This is 
where all of the usual File options are 
located (i.e. the controls for reading in or saving files, printing, etc.)

Like all spreadsheet programs, Excel gives you a workspace
that is composed of a 2D grid of ``cells'' identified by
location -- by an {\it address}.  The columns are labeled
by capital letters, and the rows are labeled by numbers.
\begin{itemize}
\item  A single cell is referenced by the
column, followed by the row, for instance {\tt B23} is
the cell in column B and row 23.
\item  A range of cells is referenced by
giving the ``starting cell,'' a colon, and the ``final
cell'' in the range.  For instance {\tt B2:B45} indicates
the cells in column B and rows 2 through 45.  {\tt B2:F2}
indicates the cells in row 2 and columns B through F.
Similarly, {\tt B2:D10} indicates all the cells in a
\emph{rectangular block} with upper left corner at cell
{\tt B2} and lower right corner at cell {\tt D10}.
\item  The addresses seen so far are all
\emph{relative addresses}.  In other words, they are
set up so that if we perform an operation in one
cell that depends on the entries to the left in its
row, then it is possible to copy and paste that operation
to other rows and the entries in the new row will be
used.  If you want to specify a \emph{fixed} address
then put in $\$$ characters:  {\tt \$C\$5} means the
cell with fixed address in column C and row 5.
(We will see several examples of this in a while;
if it is not clear why we need this distinction,
wait until you see the examples!)
\end{itemize}

The contents of a cell
can be a text label identifying what the data in a row or
column represents, a number, or a formula indicating how
to perform a desired calculation using other information
in different cells within the spreadsheet.  When you
finish entering a formula this way and press the Enter key,
the indicated computation
is performed and the result is displayed in that cell.
One \emph{very nice} feature of spreadsheets is that
if you change the contents of a cell that is used to
compute a value this way, then the calculation is
automatically performed again to update the value displayed.
We will also see this in a moment.

\subsection*{A First Worked Example}
Begin by reading in the spreadsheet {\tt First.xls}
from the course homepage.  (If you have not done so already,
download and save that in a location where you can have 
Excel read it in.)
\begin{itemize}
\item Press the ``Office Button'' at the upper
left of the Excel window (or use the File pull-down menu in Mac versions),
\item  then Open,
\item Find the appropriate location where you saved the {\tt First.xls}
file in the folder box at the top
of the Open window, highlight the file {\tt First.xls},
\item and press Open at the bottom.
\end{itemize}
\noindent
You should now see a rectangular block of cells filled
with names, text, and numbers at the upper left of the spreadsheet in rows
1 through 10 and columns A through E.  Think of this as
the grade book for a small class with 8 students (the rows are labeled
with their names) who
have had four assignments as in the labels for columns B through E.
Note that {\tt A12} has the text ``Average''  but there
are no numbers on that row (yet!).  We are going to use
Excel to compute the averages on each assignment.
\begin{itemize}
\item In cell {\tt B13}, enter the formula
{\tt $=$AVERAGE(B2:B9)}.  As you type, you will see
this showing up in the cell and in the input box
above the grid.  When you are done press Enter, and the average
will be computed and displayed.
\item  Now we will use the same method
to compute the average on each of the other assignments:
Highlight cell {\tt B13} by clicking the left mouse button
over that cell.  Make sure the Home tab at the top
of the Excel window is active, press Copy (next to the ``Office Button''),
drag the highlighting box so that all the cells in row
13, columns B to E are highlighted, and press Paste (next to Copy).
You should now see the averages for each column.
\item  In doing the averages we
were making use of the {\it relative addressing} mentioned above.
Copying the formula in one cell and pasting it into another
also changed the addresses of the cells that the formula
was applied to.  Now, we are going to perform an operation
where we want to use contents of a fixed cell on multiple
rows.  Start by filling in new information in row 14:
Put a text label ``Weights'' in {\tt A14} and the constants
$.3$ in {\tt B14}, $.25$ in {\tt C14}, $.4$ in {\tt D14}, and $.05$ in
{\tt E14}.
\item  In cell {\tt F1} add the text label ``Course Average.''
In {\tt F2} enter the formula
\centerline{\tt =\$B\$14*B2 + \$C\$14*C2 + \$D\$14*D2 + \$E\$14*E2}
You should see the weighted average displayed.
\item  You can now copy and paste that formula to the other
cells in column F and rows 3 through 9 to do the same computation
for the other students in the class.  (Note that the weights always
come from the same row, hence the fixed addresses.  Can you see what
would happen if we did not do it that way?)
\end{itemize}
Here is some other useful information:
\begin{itemize}
\item There are a number of standard mathematical functions that
can be applied to numerical contents of spreadsheet grids.  
In an Excel formula you can
square the contents of a cell by saying, for instance {\tt B13\^{}2}.
\item  If you want to take the square root of something computed from
information other cells, you 
use {\tt SQRT()}.  For instance, to compute the square root of the sum of 
{\tt B13\^{}2} and 
{\tt C13\^{}2}, you could enter\\
\centerline{\tt = SQRT(B13\^{}2 + C13\^{}2)}
in another cell of the spreadsheet.  When you press Enter, the value
will be computed and displayed in that cell.
\end{itemize}
\vfil\eject
\subsection*{The Data}
\bigskip
The country-by-country population information in Table~\ref{UNData} (on the next page) comes from Table 3 of the 2015 United Nations Demographic
Yearbook\footnote{{\tt https://unstats.un.org/unsd/demographic/products/dyb/dybsets/2015.pdf}, 
accessed on June 26, 2017.}.  Notes:
\begin{enumerate}
\item Only a selection of the roughly 200 nations are listed(!) The first seven nations are
located in Africa, the next seven are in North and South America (including the Caribbean), 
the seven following those are in Europe, and the last seven are in Oceania and Asia.  
\item The male and female populations are according to the most recent available national 
census (which varies by year according to the nation), and are in units of 1000s.
\item The 2010 and 2015 midyear populations are estimates given in units of 1000s
\item The surface area is in square km. 
\item Unavailable data is marked by a $*$
\item GDP is the market value of all final goods and services produced in a year. 
The given figures are estimates for 2015, in units of trillions of U.S. dollars.
\end{enumerate}

\subsection*{Questions}
Create a spreadsheet, enter this data, then make the following computations.
Answer the questions below in a separate text file or document.
\begin{enumerate}
\item[(A)] For each of these 28 nations, compute the male population as
a percentage of the total population according to the most recent national census. 
In which nations is the male percentage greatest?  In which is it smallest?
Why does it work out this way?
\item[(B)]  For each of the 28 nations, compute the percentage difference
between the male and female populations.
\item[(C)]  Compute the \emph{percentage change} in the populations of each of these
nations between 2010 and 2015.  (You won't be able to do this in two cases because
of missing data; just ignore those nations for this question.)
How many of these nations are losing population,
and which has the largest percentage decrease?  Which has the largest percentage
increase?  
\item[(D)]  Compute the population densities in people per square kilometer
for each of these nations.  Use the 2010 mid-year estimates, since some
data is unavailable for 2015.  Which nations have the largest population
densities among the 28 listed here?  (Note: the highest population densities
actually tend to occur for smaller, especially island, nations).
\item[(E)]  What are the 5 richest and 5 poorest of these countries, and how
should you make a fair comparison?  
\item[(F)]  Recall that the GDP figures are estimates for 2015.
Assuming that the per capita GDP was unchanged from 2010 to 2015 (that is that the 
proportion of GDP to population was the same both years), estimate
the GDP figures for each of the countries in 2010.  
\item[(G)]  What proportion of the total estimated 2010 population of the 
world, $6.9$ billion, is accounted for in the 28 countries listed in Table~\ref{UNData}?
\end{enumerate}

\begin{center}
\begin{table}[ht]
\caption{Extracts from 2015 U.N. Demographic Yearbook and 2015 GDP Figures}
\label{UNData}
\renewcommand\arraystretch{1.3}
\noindent\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
{\rm Nation} & {\rm Male} & {\rm Female} & {\rm 2010\ Mid.} & {\rm 2015\ Mid.} & {\rm Area} & {\rm GDP}\\
\hline
{\rm Egypt} & 37,219 & 35,579 & 78,685 & 88,958 & 1,002,000& .282 \\
\hline
{\rm Kenya} & 19,192 & 19,418 & 40,406 & 45,509 & 591,958 & .061\\
\hline
{\rm Mozambique}& 9,747 & 10,506 & 22,417 & 25,728 & 799,380 & .017 \\
\hline
{\rm Namibia}& 1,022 & 1,091 & 2,143 & 2,281 & 824,116 & .013 \\
\hline
{\rm Niger }& 8,519 & 8,620 & 15,204 & 19,125 & 1,267,000 & .008 \\
\hline
{\rm Nigeria}& 71,345 & 69,086 & 159,619 & * & 923,768 & .568\\
\hline
{\rm Senegal} & 6,428 & 6,445 & 12,509 & 14,357 & 196,712 & .016 \\
\hline
{\rm Argentina} & 19,524 & 20,593 & 40,788 & 43,137 & 2,780,400 & .543\\
\hline
{\rm Brazil} & 93,407 & 97,349 & 195,498 & 204,451 & 8,515,767 & 1.77\\
\hline
{\rm Canada} & 16,414 & 17,062 & 34,005 & 35,849 & 9,984,670 & 1.55\\
\hline
{\rm Chile} & 7,448 & 7,669 & 17,094 & 18,006 & 756,102 & .258\\
\hline
{\rm Dominican\ Rep.} & 4,739 & 4,706 & 9,479 & 9,980 & 48,671 & .064 \\
\hline
{\rm Honduras}& 4,052 & 4,251 & 8,046 & 8,577 & 112,492 & .019 \\
\hline
{\rm Mexico } & 54,855 & 57,481 & 114,256 & 121,006 & 1,964,375 & 1.14\\
\hline
{\rm U.S.} & 151,781 & 156,964 & 309,347 & 321,419 & 9,833,517 & 18.03\\
\hline
{\rm Austria} & 4,094 & 4,308 & 8,361 & 8,576 & 83,871 & .437 \\
\hline
{\rm Croatia} & 2,066 & 2,219 & 4,295 & 4,225 & 56,594 & .057 \\
\hline
{\rm France} & 29,715 & 31,685 & 62,918 & 64,395 & 551,500 & 2.42\\
\hline
{\rm Germany} & 39,146 & 41,074 & 81,757 & 81,198 & 357,376 & 3.36 \\
\hline
{\rm Greece} & 5,303 & 5,513 & 11,121 & 10,858 & 131,957 & .238 \\
\hline
{\rm Poland} & 18,420 & 19,624 & 38,517 & 38,006 & 312,679 & .545\\
\hline
{\rm U.K.} & 31,126 & 32,254 & 62,759 & 64,875 & 242,495 & 2.86\\
\hline
{\rm Armenia} & 1,347 & 1,525 & 3,256 & 3,011 & 29,743 & .011 \\
\hline
{\rm Australia} & 10,737 & 10,990 & 22,032 & 23,778 & 7,692,024 & 1.23\\
\hline
{\rm China} & 686,853 & 652,872 & 1,337,700 & 1,371,220 & 9,600,000 & 11.16 \\
\hline
{\rm India} & 623,270 & 587,585 & 1,182,105 & * & 3,287,263 & 2.12\\
\hline
{\rm Iran} & 37,906 & 37,244 & 74,340 & 78,773 & 1,628,750 & .425\\
\hline
{\rm Japan} & 61,829 & 65,281 & 128,070 & 126,958 & 377,930 & 4.38 \\
\hline
{\rm Pakistan} & 67,840 & 62,739 & 173,510 & 191,710 & 796,095 & .251\\
\hline
{\rm Saudi\ Arabia} & 15,307 & 11,830 & 27,563 & 31,016 & 2,206,714 & .653\\
\hline
\end{array}
\]
\end{table}
\end{center}


\begin{xcb}{Chapter Exercises}
\begin{enumerate}

\item 
In the Chapter Project for Chapter 1, we said that the freshwater ice
in the Greenland ice sheet constituted about $\frac{1}{9}$ of the world's
freshwater ice.  What percentage of the world's freshwater ice is this?
\vskip 5pt
\item 
The total population of the U.S. in 2010 was $308,745,538$
according to the Federal Census.  The 2017 population is estimated to be $324,700,000$.
\vskip 5pt
\begin{enumerate}
\item What was the percentage change in the population from 2010 to 2017?  \vskip 5pt
\item What was the percentage change per year?
\end{enumerate}
\vskip 5pt
\item 
This exercise investigates the concept of the percentage relative
error from Equation~\eqref{eq:PercentRelativeError}.
\vskip 5pt
\begin{enumerate}
\item Using a meter stick, a student measures the distance from the floor to the ceiling
in a room and finds a distance of $3.44$ meters.  The exact floor
to ceiling distance is known to be $3.40$ meters.  A second student
measures the height of a large book using a ruler and finds a length of $.44$ meters.
The actual height is $.40$ meters.  What are the \emph{absolute} and \emph{percentage
relative errors} of these measurements? \vskip 5pt
\item Which student was more \emph{accurate} in the measurement? \vskip 5pt
\item State in general how you might use the absolute and/or percentage relative
errors of measurements to assess their accuracy.  \vskip 5pt
\item Is {\it precision} of a set of measurements the same as {\it accuracy}?
Explain with an example.  (Note:  You may need to look this up.  If you do, please
cite your source!)  \vskip 5pt
\end{enumerate}
\vskip 5pt
\item 
From the web site {\tt www.campusexplorer.com}, accessed June 22, 2017:
\begin{quotation}
``The ratio of men to women in college has been slowly and steadily shifting for decades. 
More women have enrolled in college since 1979 and the ratio seems to have only recently 
stabilized at around $57\%$. That means that men make up only $43\%$ of the national student 
population.''
\end{quotation}
\vskip 5pt
\begin{enumerate}
\item Does this statement make sense mathematically?  Criticize from the point of view
of the concepts introduced in this chapter.  \vskip 5pt
\item What should the correct value of the ratio of men to women be if, 
as the authors apparently intended to say, 
women make up $57\%$ of the student body and men make up $43\%$ of the student body?
\end{enumerate}
\vskip 5pt
\item
\label{ex:Solar}
According to data from the Solar Energy Industries Association\footnote{
{\tt http://www.seia.org/research-resources/top-10-solar-states}, accessed on June 22, 2017.}, 
the ``top 10 solar states'' in terms of (cumulative) total solar electric generating capacity
(measured in megawatts, Mw) installed in solar panel farms, homes, etc. as of 2016 are
given in Table~\ref{StateSolarTable}.
\begin{table}[ht]
\caption{Table of Solar Electric Generating Capacity by State}
\label{StateSolarTable}
\renewcommand\arraystretch{1.5}
\noindent\[
\begin{array}{|c|c|c|c|}
\hline
{\rm State} & {\rm Capacity\ in\ Mw} & {\rm Population} & {\rm Capacity\ Per\ Capita} \\
\hline
{\rm California}& 18,296 & & \\
\hline
{\rm North\ Carolina}& 3,016 & & \\
\hline
{\rm Arizona}& 2,982 & & \\
\hline
{\rm Nevada}& 2,191 & & \\
\hline
{\rm New\ Jersey} & 1,991 & & \\
\hline
{\rm Utah} & 1,489 & & \\
\hline
{\rm Masschusetts}& 1,487 & & \\
\hline
{\rm Georgia }& 1,432 & & \\
\hline
{\rm Texas} & 1,215 & & \\
\hline
{\rm New\ York} & 927 & & \\
\hline
\end{array}
\]
\end{table}
\vskip 5pt
\begin{enumerate}
\item Look up estimates for the populations of these states in 2016, fill in that
column of the table, then use that data to compute the \emph{per capita solar electric
capacity} for each state and fill in the last column.  \vskip 5pt
\item Discuss any interesting patterns you see here.  In particular, if you 
make the comparison on the per capita basis, are the states ordered the same way or
do some states move higher or lower in the rankings?
\end{enumerate}
\vskip 5pt
\item  
\label{ex:CityPDs}
In Table~\ref{CityTable}, we give the populations and geographic areas of some
of the larger U.S. cities according to the 2010 Federal Census.  Notes: The populations and areas
are given only for the incorporated cities, not the surrounding metropolitan areas. 
The figures for New York City are different from those given in the text because
those were estimates for the year 2016.
\begin{table}[ht]
\caption{Table of City Populations and Areas in 2010}
\label{CityTable}
\renewcommand\arraystretch{1.5}
\noindent\[
\begin{array}{|c|c|c|}
\hline
{\rm City} & {\rm Population} & {\rm Land\ Area\ (in\ square\ miles) } \\
\hline
{\rm Boston}& 645,149 & 48.43\\
\hline
{\rm Chicago}& 2,695,598& 227.13\\
\hline
{\rm Miami}& 399,457& 35.67\\
\hline
{\rm New\ York\ City}& 8,175,133 & 302.6\\
\hline
{\rm Philadelphia} &1,517,550 &135.09 \\
\hline
{\rm San\ Francisco} &805,235 &46.69\\
\hline
\end{array}
\]
\end{table}
\begin{enumerate}
\item Compute the average population densities for all of these cities
in units of people/square mile, then people/square kilometer.  \vskip 5pt
\item What was the \emph{percentage change} in the average population density
of New York City between 2010 and 2016?  Does it make any difference
which set of population density figures (i.e. the two in units of people/square mile or 
the two in units of people/square kilometer) you use here?  Explain.
\end{enumerate}
\vskip 5pt
\item
\label{ex:NYBoroughPDs}
In Table~\ref{BoroughTable}, we give the populations and 
land areas of the five boroughs of New York City, again according to the 
2010 Federal Census.\footnote{You should 
check that they add up to the
total given for the whole city in Exercise~\ref{ex:CityPDs}.  
The total reported for Manhattan on the Wikipedia page  
{\tt https://en.wikipedia.org/wiki/List$\_$of$\_$United$\_$States$\_$cities$\_$by$\_$population$\_$density} 
is different for some reason that I do not understand.  The figure there is either
a simple transcription error, or it may also not include populations like that of Roosevelt Island, which is 
administratively part of the borough of Manhattan but not contiguous with Manhattan Island.}  
\begin{table}[ht]
\renewcommand\arraystretch{1.5}
\caption{Table of New York City Borough Populations and Areas in 2010}
\label{BoroughTable}
\noindent\[
\begin{array}{|c|c|c|}
\hline
{\rm Borough} & {\rm Population} & {\rm Land\ Area\ (in\ square\ miles) } \\
\hline
{\rm Bronx}& 1,385,108 & 42.10 \\
\hline
{\rm Brooklyn}& 2,504,700 & 70.82\\
\hline
{\rm Manhattan}& 1,585,873 & 22.83\\
\hline
{\rm Queens}& 2,230,722 & 108.53\\
\hline
{\rm Staten\ Island} & 468,730 & 58.37 \\
\hline
\end{array}
\]
\end{table}
\vskip 5pt
\begin{enumerate}
\item Compute the average population densities for all of these
in units of people per square mile, then people per square kilometer.  \vskip 5pt
\item Is the numerical average of the 5 borough population densities
the same as the overall New York City population density from 
Exercise~\ref{ex:CityPDs}?  Why or why not?  If you knew the 
five borough population densities from this problem, what could 
you do to recover the city population density?
\end{enumerate}
\vskip 5pt
\item
Refer to Table~\ref{MercuryInFish} in Example~\ref{Fish}.
\vskip 5pt
\begin{enumerate}
\item How much mercury (in grams) would we expect to find in a 25\ kg sample of
grouper flesh?   \vskip 5pt
\item What is the percentage difference between the mercury level reported for white (albacore)
tuna and light (skipjack) tuna?  Based on that proportion, how often do you expect
the EPA recommends that children and adults can eat light tuna?  Check this online.  \vskip 5pt
\item What is the percentage difference between the mercury level reported for swordfish
and shrimp?  Why should it be true that mercury levels are so much higher in swordfish than
in shrimp?  If you need to, look up the feeding habits of these animals.  \vskip 5pt
\item  What is the percentage difference between the mercury level reported for lobster
and scallops?  Why should it be true that mercury levels are so much higher in lobster than
in scallops?  If you need to, look up the feeding habits of these animals.  
\end{enumerate}
\vskip 5pt
\item
(Hypothetical; this is not real data.)
A scientist sends some samples of water contaminated with lead to a commercial lab for 
analysis.  To check the accuracy of its measurements, the lab first does a calibration 
run on a sample known to have a concentration of $150$ ppb (parts per billion).  
The lab's machine returns a measured concentation of $159$ ppb. \vskip 5pt
\begin{enumerate}
\item What is the percentage relative error in this measured concentration?  \vskip 5pt
\item One of the scientist's samples yields a value of $430$ ppb when the lab analyzes it. 
Assuming that measurement has the same percentage relative error
as the calibration run, estimate the actual lead concentration in ppb in that sample.  \vskip 5pt
\item Discuss whether the procedure described in part (b) is necessarily justified.
(For instance, what if the machine was just mis-calibrated and all the values it returned
were biased by the same absolute amount?)
\end{enumerate}  
\vskip 5pt
\item
Refer to Example~\ref{TwoDice}.
\vskip 5pt
\begin{enumerate}
\item Show that the probabilities for the ``loaded'' dice given in Equations~\eqref{eq:Loaded1}, \eqref{eq:Loaded2}, 
and \eqref{eq:Loaded3} add up to 1 if we sum over all the elements of $S$.  (Hint: You will need to 
determine how many outcomes there are in each case.)\vskip 5pt
\item If all pairs $(m,n)$ are equally likely, what is the probability of the event $E = $ the numbers
of dots sum to $7$?\vskip 5pt
\item What is the probability of the event $E = $ number of black dots even and number of white dots
odd in the loaded case from part (a)?
\end{enumerate}

\vskip 5pt
\item
The severity of hurricanes, floods and other disasters are often described in terms
related to probabilities.  For instance a ``100-year hurricane'' is a hurricane that
on average would only be expected to occur (only) once in a 100-year period, while
the other hurricanes in that period are less severe.  
\vskip 5pt
\begin{enumerate}
\item How many 100-year hurricanes would be expected to occur in a 500-year period?
In a 50-year period?  In a 1-year period?  Explain. (Hint: numbers between $0$ and $1$ 
interpreted as probabilities are OK.)  \vskip 5pt
\item Which would be expected to be more destructive -- a 100-year hurricane or
a 500-year hurricane?  Explain.
\end{enumerate}

\vskip 5pt
\item
Express each of the following statements as an equation by using 
proportionality.  (See Example~\ref{CircleAreaRadius}.)\vskip 5pt
\begin{enumerate}
\item The volume of a sphere is proportional to the cube of its radius.\vskip 5pt
\item The volume of a right circular cylinder is proportional to the product of the square
of the radius and the height.\vskip 5pt
\item The volume of a right circular cone is proportional to the product of the square
of the radius and the height.\vskip 5pt
\item ``Newton's Gravitation Law:'' The gravitational force exerted by one body 
on another is proportional to the reciprocal of the square of the distance between them.
(The proportionality factor depends on the two masses.) \vskip 5pt
\end{enumerate}

\vskip 5pt
\item
Refer to Equation~\eqref{eq:CaptureRecapture}.  \vskip 5pt
\begin{enumerate}
\item Give another explanation why this method works (approximately) by interpreting the ratios
as probabilities.\vskip 5pt
\item Which situation gives a \emph{larger estimate for the total population}:  
\begin{itemize} 
\item a large proportion of 
the individuals in the second sample are found to be tagged, \emph{or} 
\item a small proportion of the 
individuals in the second sample are found to be tagged?
\end{itemize}
Explain.  \vskip 5pt
\item A random sample of 500 mosquitoes in a neighborhood is captured, tagged, and released.  A few hours later, 
a second sample of 400 is taken and the sample contains 10 of the tagged mosquitoes.  What is the estimated
population of mosquitoes?
\end{enumerate}

\end{enumerate}
\end{xcb}

\endinput
