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\centerline{MONT 107N -- Understanding Randomness}
\centerline{Group Discussion 1}
\centerline{January 22, 2010}
\vskip 10pt
\noindent
{\it Background and Goals}
\vskip 10pt
\noindent
We have discussed ``box models'' as a way to describe
chance processes involving repetition.  Recall the 
key components of a box model are:
\begin{itemize}
\item the numbers that go into the box
\item how many of each kind there are
\item how many draws are made from the box and whether the draws
are done with or without replacement
\end{itemize}
Usually, the outcomes from the chance process will correspond
to the possible {\it sums} of the numbers on the tickets that we draw
from the box.  Today, we want to practice
with several box models of this type and start to understand
the random processes they describe.   
You will be handing in one set of 
solutions from each group no later than Monday, January 25.
(Of course if you are done by the end of class, you can hand
your solutions in then, too!)
\vskip 10pt
\noindent
\begin{enumerate}
\item[A.]  Suppose that we do 1000 random draws with replacement
from a box containing exactly three tickets with the numbers $2,5,8$.
\begin{enumerate}
\item[1.]  What is the {\it smallest} sum we could obtain?  What
is the {\it largest} possible sum?  Explain your answers.
\item[2.]  Explain why the sum of the numbers you draw is 
probably close to 
$$ 1000 \times \text{average of the box} = 5000 $$
(Hint:  What are the chances of drawing each different number?  
About how many times would you expect to draw each in 1000 draws?)
\end{enumerate}
\item[B.]  Consider the Nevada version of the casino 
game roulette as described on Wednesday.  One bet you can 
place in roulette is a wager on {\it four adjoining numbers} 
on the table -- a collection of four numbers like 11,12,14,15 
whose squares share a corner in the grid (see diagram on page 
282 of the text).  A four numbers bet pays 8 to 1. 
\begin{enumerate}
\item[1.]   Set up and describe a box model that is equivalent 
to betting repeatedly on a particular collection of 4 numbers 
(specify the features given in the Background and Goals section 
above).  
\item[2.]  What are the average and SD of the numbers
in your box from part 1?
\item[3.]  If you draw 100 times from your box as in part 1,
explain why the sum of the numbers you draw is 
probably close to 
$$ 100 \times \text{average of the box} $$
(Hint:  What are the chances of drawing each different number?  
About how many times would you expect to draw each in 100 draws?)
\item[4.]  The precise version of the ``law of averages''
will say that if you draw $n$ times from the box,
the chance error -- the difference between the sum of the numbers
you actually draw and $n \times \text{average of box}$ -- 
will almost always be at most $2 \sqrt{n} \times \text{SD of box}$
in absolute value:
$$|\text{chance error}| \le 2 \sqrt{n} \times \text{SD of box}.$$
What does this say about the relative error:
$$\text{chance error}/n$$
as $n$ gets larger and larger?
\end{enumerate}
\item[C.]  In roulette, you can also place a bet on any one of the 
columns $1,4,7,\ldots,34$, $2,5,8,\ldots,35$, or $3,6,9,\ldots,38$
(again, see page 282 in the text).  
A column bet pays 2 to 1.   
\begin{enumerate}
\item[1.]  Set up and describe a 
box model that is equivalent to betting repeatedly on 
a particular column (again, specify the features 
given in the Background and Goals section above).
\item[2.]  What are the average and SD of the numbers
in your box from part 1?  
\item[3.]  If you draw 100 times from your box as in part 1,
explain why the sum of the numbers you draw is 
probably close to 
$$ 100 \times \text{average of the box} $$
\item[4.]  The precise version of the ``law of averages''
will say that if you draw $n$ times from the box,
the chance error -- the difference between the sum of the numbers
you actually draw and $n \times \text{average of box}$ -- 
will almost always be at most $2 \sqrt{n} \times \text{SD of box}$
in absolute value
$$|\text{chance error}| \le 2 \sqrt{n} \times \text{SD of box}.$$
What does this say about the relative error:
$$\text{chance error}/n$$
as $n$ gets larger and larger?
\end{enumerate}
\end{enumerate}
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