% LaTex2e
\documentclass[12pt]{article}
\usepackage{amssymb,latexsym}

%%%%%%% \usepackage{anysize}
%\marginsize{left}{right}{top}{bottom}
%\marginsize{1.25cm}{1.25cm}{1.75cm}{2.5cm} % this cuts off page numbers
%%%%%% \marginsize{2cm}{2cm}{1cm}{3cm}

\pagestyle{myheadings}
\def\header{{\hfill Exam 1 \hfill A. Miller \hfill Fall 2008
                                       \hfill Math 210\hfill}}
\markboth\header\header

\newcount\probno
\newcount\total
\newwrite\examTBLFile

\probno=0

\def\prob#1{\advance\probno by 1 \par  \bigskip    \vfill \newpage
    \noindent\the\probno . (#1 pts)\advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}

\def\finishtbl{
    \immediate\write\examTBLFile{Total \string& \the\total
    \string& \qquad \string\\ \string\hline}
     }

\newwrite\ans
\immediate\openout\ans=answers
\outer\def\answer{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\answerx{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\putanswer{\par\medbreak
  \immediate\write\ans{}
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copyans}
\def\setupcopy{\def\do##1{\catcode`##1=12}\dospecials
     \catcode`\|=12\obeylines}
{\obeylines \gdef\copyans#1
  {\def\next{#1}%
  \ifx\next\empty\let\next=\endgroup%
  \else\immediate\write\ans{\next}\let\next=\copyans\fi\next}}
\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

% \def\answer{\par\bigskip\bigskip\bigskip\noindent Answer: \par}
% \outer\def\answer{ {\par\bigskip Circle your answer.} \par\medbreak

\newenvironment{enum}
{\begin{list}{(\alph{enumi})}
{\usecounter{enumi}\setlength{\rightmargin}{\leftmargin}}}
{\end{list}}

\newenvironment{Enum}
{\begin{list}{(\Alph{enumi})}
{\usecounter{enumi}\setlength{\rightmargin}{\leftmargin}}}
{\end{list}}

\def\pr{\prime}
\def\vs{\vspace{1in}}

\begin{document}

\bigskip

No notes, no books, no calculators, no cell phones, no pagers, no
electronic devices of any kind.


\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\bigskip 
Circle your TAs name:

\bigskip

\hfill Song Sun \hfill
(Boyd) Chalermpong Worawannotai \hfill

\bigskip
Hand in your exam to your TA.

\bigskip
\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
   \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

\bigskip

Solutions will be posted shortly after the exam:

www.math.wisc.edu/$\sim$miller/m210

\setcounter{page}{0}
\newpage



\prob{10} % 1.1-31
Suppose $$A\times B=\{(a,b),(a,x),(b,b),(b,x)\}$$ and
$$C=\{0,1\}.$$  What are the following sets:

\bigskip\bigskip

$A=$

\vs

$B=$ 

\vs

$A\times C=$

\answer $A=\{a,b\}$, $B=\{b,x\}$, $A\times C=\{(a,0),(a,1),(b,0),(b,1)\}$

\prob{10} % 1.2 - 17
Let the universal set be $U=\{1,2,3,4,5,6,7,8\}$ and let
$A=\{2,6\}$ and $B=\{1,2,4\}$.  What are the following sets:

\vs

$A^\prime\cap B=$

\vs

$A\cup B=$

\vs
$A\cap B^\prime=$


\answer $\{1,4\}$,  $\{1,2,4,6\}$,  $\{6\}$ 

\prob{10} %12.2 -  5
Construct the truth table for
$$(p\wedge q)\vee ((\sim p) \;\wedge\; (\sim q)) $$


\answer The TT and FF lines are true.

\prob{10} % 1.3 - 27
Sets $A$, $B$, and $C$ are subsets of the universal set $U$. 
Suppose $n(U)=80$, $n(A)=15$, $n(A\cap B)=5$, $n(C)= 30$, and
$n(A\cup B^\prime)=55$, and $(A\cup B)\cap C=\emptyset$.

Find the following numbers:

\bigskip

$n(C^\prime)=$

\vs

$n(A\cap C)=$

\vs

$n(A^\prime\cap B)=$

\answer 50, 0, 25


\prob{10} % 2.2 - 25
A product code consists of a string of 3 different letters
from the set $\{A,B,C,D,E,F\}$.  For example, ADE or DEB.

How many different
product codes are there which contain the letter $E$ but do not
contain the letter $F$?

\answer 36



\prob{10} % 2.3 - 23 2.4 -  5
A club has 3 girls and 5 boys.  A group of 3 is to
be selected.  Assume that each group of 3 is equally likely to
be selected.
What is the probability that the selected group contains at least
one boy and at least one girl?

\answer $45/56$


\prob{10} % 3.1 - 27
Suppose $A,B,C$ are events in a probability space $S$.  Suppose
that $A\cap B$, $B\cap C$, $A\cap C$ is a partition of $S$.  If
$Pr[A]=.5$ and $Pr[B]=.6$, then what is $Pr[C]$?


\answer $.9$


\prob{10} % 3.2 -  9
Let $A$ and $B$ be events such that $Pr[A]=.3$ and
$Pr[B]=.6$. 

\bigskip What is $Pr[A\cup B]$ if
$A$ and $B$ are disjoint?

\vs What is $Pr[A\cup B]$ if
$A$ and $B$ are independent?

\answer $.9$, $.72$


\prob{10} % 3.4 -  21
Tamara has two cell phones, brand A and brand B.
She uses brand A 20 percent of the time and brand B
80 percent of time.  She notices that she losses
the signal with brand A 20 percent of the time and
with brand B 10 percent of the time.   On a
random use of one of her cell phones she losses the
signal.  What is the probability she was using
brand A?

\answer $1/3$


\prob{10} % 3.5 - 31
A multiple-choice exam consists of 5 questions.  Each question has 
4 possible answers, only one being correct.  A student
randomly guesses each answer.  Assume this situation is modeled by
Bernoulli trials.

\bigskip
What is the probability that the
student gets exactly 3 correct answers?

\answer $45/512$

\printanswers

\finishtbl

\end{document}



