\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\bx{\boldsymbol{x}}
\def\by{\boldsymbol{y}}
\def\v{\boldsymbol{v}}
\def\bb{\boldsymbol{b}}
\def\be{\boldsymbol{e}}
\def\A{{\boldsymbol A}}
\def\B{{\boldsymbol B}}
\def\I{{\boldsymbol I}}

\begin{document}

\title{Finite Mathematics}
\author{Final Exam}
\date{May 27, 2020, 16:00--18:00}
\maketitle

\begin{enumerate}

\item Are the following vectors linearly independent?  If not,
determine a maximal set of linearly independent vectors, and express
the other vectors as linear combinations of vectors from this set.
\[
  \v_1 =
  \begin{pmatrix}
    -1 \\ -1 \\ 5
  \end{pmatrix} \,,
  \quad
  \v_2 =
  \begin{pmatrix}
    2 \\ 2 \\ -10
  \end{pmatrix} \,,
  \quad
  \v_3 =
  \begin{pmatrix}
    -1 \\ 2 \\ 1
  \end{pmatrix} \,,
  \quad
  \v_4 =
  \begin{pmatrix}
    0 \\ 3 \\ -4
  \end{pmatrix} \,.
\]
\hfill(10)

\item Find the best fit of the equation of a straight line,
$y=m x + b$, in the least-square sense to the uncertain
measurements $y_1=1$, $y_2=3$, $y_3=3$ that correspond to the given,
certain values $x_1=0$, $x_2=1$, $x_3=2$.
\hfill(10)


\item Five distinct points lie on the rim of a circle. Choosing the
points as vertices, how many different triangles can be drawn?
\hfill(5)

\item Sarah is traveling home on Deutsche Bahn.  Official statistics
say that at Hannover, there is a 10\% chance that the scheduled
connection will be missed, at Berlin there is a 20\% chance the
scheduled connection will be missed.  From past experience, she knows
that she won't arrive on the scheduled train home in one out of four
trips.

Is the probability of missing the connection in Berlin independent of
the probability of missing the connection in Hannover?  If not, what
is the probability of missing the connection in Berlin given that she
made her connection in Hannover on time?
\hfill(5)

\item An urn contains two red balls and three green balls.  A ball is
drawn and replaced.  In five such trials, what is the probability that
exactly three will come out red?
\hfill(5)

\item A travel insurance policy provides cancellation cover and
emergency return for trips abroad.  Past data suggests that 6\% of
trips are canceled and eligible for a reclaim of the ticket price.
In 1\% of trips, the traveler is eligible for an emergency return at
an average cost of three times the original ticket price.  What
percentage of the ticket price must be charged for insurance cover,
neglecting administrative expenses, so that the insurance company
breaks even?
\hfill(5)

\item A student is taking a multiple choice exam with four given
answers per question.  On half of the questions, one given answer is
so implausible that it can be eliminated without having studied at
all, the other answers are equally plausible unless the student knows
the correct answer.  The student has studied just enough to know the
correct answer to half of the questions.

If the student gives a correct answer, what is the probability that
they knew the answer?
\hbox{~~~~}\hfill(10)

\item The weather on a given day is classified as dry or rainy.  It is
observed that ``one out of two rainy days is followed by a dry day,
and one of out four dry days is followed by a rainy day''.

\begin{enumerate}
\item Find a Markov chain that is consistent with this observation.

\item Assuming this model, what is the probability that, given that a
day is rainy, the next three days will also be rainy?

\item Assuming this model, find the long-term proportion of rainy
days.

\item Is a Markov chain the only possible model for the given
observation?  Explain!
\end{enumerate}
\hfill(5+5+5+5)


\item An urn contains three red balls and three green balls.  A ball
is chosen at random without bias.  If it is red, it is removed.  If it
is green, it is replaced.  This process is repeated until all red
balls are removed.

\begin{enumerate}
\item Show that this process is an absorbing Markov chain by writing
out its transition matrix in canonical form.

\item What is the expected number of steps before the process stops?
\end{enumerate}
\hfill(5+5)


\item Find the optimal strategies and the value of the game for the
two-player zero-sum games represented by the following pay-off
matrices:

\begin{enumerate}
\item $\displaystyle G = \begin{pmatrix}
1 & -2 \\ -1 & 2
\end{pmatrix}$
\item $\displaystyle G = \begin{pmatrix}
2 & -2 & 1 & -1 \\
1 & 2 & 0 & 0 \\
4 & 2 & -2 & -4 \\
2 & 1 & -1 & -2 
\end{pmatrix}$
\end{enumerate}
\hfill(5+5)
\end{enumerate}
\end{document}

