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\begin{document}

\centerline{A. Miller \hfill M431 \hfill Exam 1 \hfill Spring 98}
\begin{center}
  Do not hand in this sheet. Show all work.
\end{center}

% chapt 1
%20
\prob A person has 9 friends of whom 5 will be invited to a party.
\par (a) How many choices are there if 2 of the friends are feuding
and will not attend together?
\par (b) How many choices are there if 2 will only attend together?
\answer (a) $\ch{7}{5}+2\ch{7}{4}=91$ (b) $\ch{7}{5}+\ch{7}{3}=56$

%28
\prob 24 new teaching positions are to be divided among 6 schools.
\par (a) How many divisions are possible? (For example, giving
4 positions to each school counts as one division.  Another
division is to give all 24 positions to the first school.)
\par (b) How many division are there so that each school gets
at least two new teachers?
\answer (a) $\ch{29}{5}=118755$ (b) $\ch{17}{5}=6188$

% chapter 2
\prob  A bridge hand consists of 13 cards.  A Yarborough
is a hand which does not contain a card which is a ten or
higher.  Ten or higher means Ten, Jack, Queen, King, or Ace of
any suit.  Many friendly bridge games follow the rule that if
one of the players gets a Yarborough, then all the hands are thrown in
and redealt.
\par What is the probability that a bridge hand is a not a
Yarborough?
\answer $1-{\ch{32}{13}\over \ch{52}{13}}$

%54

\prob A bridge hand is void in clubs if it
does not contain any clubs.  There are four suits: clubs, diamonds,
hearts, and spades; and each has $13$ cards.
\par What is the probability that a bridge hand
is void in at least one suit?
\answer $4{\ch{39}{13}\over \ch{52}{13}}-6{\ch{26}{13}\over \ch{52}{13}}
+4{1\over \ch{52}{13}}=.0519$


% chapter 3

%8 from chapt 2
\prob Suppose $A$ and $B$ are two events such that
$P(A)=.2$, $P(B)=.7$ and $P(A|B)=.05$
\par\noindent What is the probability that
\par (a) both $A$ and $B$ occur
\par (b) either $A$ or $B$ occurs
\par (c) $A$ occurs but $B$ does not
\par (d) $P(B|A)$.
\answer (a) $.035$ (b) $.865$ (a) $.165$ (a) $.175$

%45
\prob  Suppose that a cancer diagnostic test is 95\% accurate for
those that have the disease, but only 90\% accurate for those that
do not.   Suppose that 1\% of the population have the disease.
What is the probability that a person who tests positive (for cancer)
has the disease?
\answer $.0876$

% chapter 4
% 25 from chapt 3
\prob All the workers at a certain
company drive to work and park
in the company's lot.  All the cars contain either one, two, or
three workers.  About half the workers
drive solo, i.e., their car only contains them.
The expected value of the number of workers in each
car is $1.6$.  What is the probability that a random car has three
workers in it?
\answer $.1$

% 54
\prob Suppose that the average number of cars abandoned weekly
on a certain highway is $4$.  Use a Poisson distribution to
approximate the probability that there will be
\par (a) no abandoned cars in the next week;
\par (b) at least $2$ abandoned cars over the next $3$ weeks.
\answer (a) $1.8315639$ (b) $.99992013$


\printanswers

\end{document}