\documentstyle[12pt]{article}
\pagestyle{empty}

% save paper style
%\oddsidemargin 5pt \evensidemargin 5pt \marginparwidth 68pt
%\marginparsep 10pt \topmargin -10pt \headheight 10pt
%\headsep 15pt \footskip 35pt
%\textheight = 625pt  \textwidth 440pt \columnsep 10pt \columnseprule 5pt

\newcount\probno
\probno=0
\def\prob{\advance\probno by 1 \bigskip\par\noindent\the\probno . }
\def\ch#1#2{\left(\begin{array}{c}#1\\#2\\ \end{array}\right)}

\newwrite\ans
\immediate\openout\ans=answers
\outer\def\answer{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\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{}

\begin{document}

\centerline{A. Miller \hfill M431 \hfill Exam 2 \hfill Spring 98}
\begin{center}
  Do not hand in this sheet. Show all work.
\end{center}

% chapt 5 , 11
\prob A point is chosen at random on a line segment of length $L$
and the line segment is then cut at that point into two subsegments.
Find the probability that the ratio of the shorter to the longer
subsegment is less than $1\over 3$.
\answer $1\over 2$

% chapt 5, 23
\prob
In the following use the normal approximation to the
binomial distribution (DeMoivre-Laplace).  You may express your
answer in terms of $\Phi$.
One thousand rolls of a fair die will be made. If the number
$2$ appears exactly $500$ times, compute the approximate
probability that the number $5$ appears between 90 and 110 times.
\answer $\Phi({10\over\sqrt{8}})-\Phi(-{10\over\sqrt{8}})\approx .7$

% chapt 5, 37
\prob If $X$ is a random variable with
the exponential distribution with parameter $\lambda=1$.
Suppose $Y=e^X$.  Find the probability that $2<Y<3$.
\answer $1\over 6$

% chapt 6, 2
\prob Suppose two balls are randomly selected without replacement
from an urn
containing 2 white and 3 red balls.  Let $W$ be the number of
white balls selected and $R$ be the number of red balls selected.
Give the joint probability mass function of $W$ and $R$.
\answer \par $P(W=2,R=0)={1\over 10}$
\par $P(W=1,R=1)={6\over 10}$
\par $P(W=0,R=2)={3\over 10}$
\par all other combinations have probability $0$.

% chapt 6, 17
\prob Three points $X,Y,Z$ are randomly selected (uniformly and
independently)
in the unit interval $[0,1]$.  What is the probability that $X<YZ$?
\answer $1\over 4$


% chapt 7
\prob An ordinary deck of playing cards is shuffled and then seven cards are
dealt out one at a time.  Let $A$ be the number of aces dealt out.
Find the expected value of $A$ and the variance of $A$.
\answer $E(A)={7\over 13}\approx .5385$
\par $var(A)=7({1\over 13})({12\over 13})+
42({1\over 13}{3\over 51}-{1\over 13}{1\over 13})\approx .4386$

% chapt 7
\prob You deal out a well-shuffled pack of
cards until you have seen five spades.  How many cards can you expect
to deal out?
\answer $5+39({5\over14})\approx 18.93$



% chapt 7
\prob A fair die is rolled until we roll a six.
Let S be the sum of the rolls.  For example, if
we roll $4,1,4,2,6$, then $S=4+1+4+2+6=17$.  Find the
expected value of $S$.
\answer $21$

\printanswers

\end{document}