\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{Makeup Final Exam}
\date{August 25, 2020, 17:00--19:00}
\maketitle

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\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.
\[
  \bx_1 =
  \begin{pmatrix}
    1\\-1\\0\\1
  \end{pmatrix} \,,
  \quad
  \bx_2 =
  \begin{pmatrix}
    1\\1\\-1\\0
  \end{pmatrix} \,,
  \quad
  \bx_3 =
  \begin{pmatrix}
    1\\ -3\\  1\\  2
  \end{pmatrix} \,,
  \quad
  \bx_4 =
  \begin{pmatrix}
    1\\  3\\ -2\\ -1
  \end{pmatrix} \,.
\]
\hfill(10)

% In [1]: a = array([1,-1,0,1])
% In [2]: b = array([1,1,-1,0])
% In [3]: 2*a-b
% Out[3]: array([ 1, -3,  1,  2])
% In [4]: 2*b-a
% Out[4]: array([ 1,  3, -2, -1])

%\newpage

\item Consider the plane with parametric representation
\[
  \bx =
  \begin{pmatrix}
    -1\\2\\0
  \end{pmatrix}
  + \lambda
  \begin{pmatrix}
    0\\1\\1
  \end{pmatrix}
  + \mu
  \begin{pmatrix}
    -1\\-3\\1
  \end{pmatrix} \,.
\]
Compute the values for the parameters $\lambda$ and $\mu$ 
which correspond to the point
in the plane closest to the given point
\[
  \boldsymbol p =
  \begin{pmatrix}
    -7\\ -4\\  0
  \end{pmatrix} \,.
\]
\hfill(10)

% In [5]: a = array([-1,2,0])
% In [6]: v = array([0,1,1])
% In [7]: w = array([-1,-3,1])
% In [8]: a-v+2*w
% Out[8]: array([-3, -5,  1])

%\newpage
\item In how many ways can $5$ people be made to stand in a straight
line? In a circle?
\hfill(5)

%\newpage
\item
For a three-child family, let the events $E$ and $F$
be as follows.
\begin{itemize}
\item[$E$:] The family has at least one boy
\item[$F$:] The family has children of both sexes
\end{itemize}
Find the following.
\begin{enumerate}
\item $P(E)$
\item $P(F)$
\item $P(E \cap F)$
\item Are $E$
and $F$ independent?
\end{enumerate}
\hfill(5)

% a 7/8
% b 6/8
% c 6/8
% d no

%\newpage
\item If a basketball player makes $3$ out of every $4$ free throws,
what is the probability that she will make $2$ out of $3$ free throws
in a game?
\hfill(5)

% The probability of making a free throw is 3/4. Therefore, p=3/4,
% q=1/4, n=3, and k=2.
% Therefore,
% b(3,2;3/4) = 3C2 (3/4)^2 (1/4)^1
% = 3*9/16 * 1/4 = 27/64

%\newpage
\item An oil drilling company has determined that it costs \$10\,000
to sink a test well. If oil is hit, the revenue for the company will
be \$500\,000. If natural gas is found, the revenue will be
\$150\,000. If the probability of hitting oil is 3\% and of hitting
gas is 6\%, how much should they be willing to pay for the licence to
drill? 
\hfill(5)

%\newpage
\item A person has two coins: a fair coin and a two-headed coin. A
coin is selected at random, and tossed. If the coin shows a head, what
is the probability that the coin is fair?
\hfill(10)

%\newpage
\item
% From One Hundred Solved Exercises for the subject:
% Stochastic Processes I
% by Takis Konstantopoulos
I have 2 umbrellas, some at home, some in the office. I keep moving between home
and office. I take an umbrella with me only if it rains. If it does not rain I leave the
umbrella behind (at home or in the office). It may happen that all umbrellas are in
one place, I am at the other, it starts raining and must leave, so I
get wet.

I live in a place where it rains half of the time.
\begin{enumerate}
\item Write out the transition matrix for a Markov chain describing
this situation.

\emph{Hint:} As states, take the number of umbrellas in the place
where I am currently at (home or office).

\item Show that this Markov chain is regular.

\item Find the stationary distribution.

\item What is the probability that I get wet on any given walk between
home and office?
\end{enumerate}
\hfill(5+5+5+5)

%\newpage \mbox{} 
%\newpage
\item An urn contains three black balls and three white balls.  A ball
is chosen at random without bias.  If it is black, it is removed.  If
it is white, it is replaced.  This process is repeated until all black
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)

%\newpage

\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 $\displaystyle G = \begin{pmatrix}
-25 & 10 \\ 10 & 50
\end{pmatrix}$
\item $\displaystyle G = \begin{pmatrix}
1 & -2 \\-3 & 4
\end{pmatrix}$
\end{enumerate}
\hfill(5+5)

%\newpage \mbox{} 
%\newpage

\end{enumerate}
\end{document}

