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\begin{document}

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

\begin{enumerate}

\item All linear algebra topics covered up to the mock midterm exam.
Note that solving a system of equations is part of the solution
process to find equilibrium properties for regular Markov chains;
inversion of a matrix is required in the context of absorbing Markov
chains.

Least square solutions and/or least norm solutions are important; you
should expect at least one of them on the final exam.

\item Basic notions from set theory, counting (multiplication
principle, permutations, combinations): important to review, may not
be tested directly, but as part of a probability problem.

(Homework 6 and Homework 7, Questions 1 and 2.)

\item Basic notions from probability: mutually exclusive events,
addition rule, conditional probability, independent events.

All the Chapter 14 ``Chapter Review'' questions from the CNX notes are
good review problems.  There will similar questions on the final exam.

\item Bernoulli trials (binomial probability), expected value.

The Chapter 15 ``Chapter Review'' questions from the CNX notes provide
plenty of practice material.

\item Bayes' formula.

\emph{Example problem:}  There are two types of students.  Students of type $A$ report
sick only when they are actually sick.  Students of type $B$ report
sick with probability $1$ whenever there is an exam.  It is known that
the sickness rate of the general working population is $5\%$.  On
every exam, $10\%$ of students report sick.
\begin{enumerate}
\item  What is the probability that a randomly selected student
is of type $B$?
\item What is the probability that a student who reports sick on
the day of the exam is of type $B$?
\item A student reports sick on the midterm and on the final.
What is the probability that the student is of type $B$?  
\end{enumerate}

(This is an old exam problem from the final exam of ``Engineering and
Science Mathematics 2B'' from Spring 2013.  You can find a worked-out
answer on my web page.  Problem 6 from that same final exam provides
another nice Bayes' rule problem.)

\item Regular Markov chains: Check whether a Markov chain is regular.
Find the vector of equilibrium probabilities by solving a linear
system of equations.

\emph{Example problem:} A truck driver distributes goods between
warehouses in Amsterdam, Berlin, and Copenhagen.  For a tour starting
in Amsterdam, there is a 30\% chance that it'll end in Amsterdam
again, a 30\% chance that it'll end in Berlin, and a 40\% chance that
it'll end in Copenhagen.  For a tour starting in Berlin, there is a
40\% chance that it'll go to Amsterdam, a 40\% chance that it'll
remain in Berlin, and a 20\% chance that it'll go to Copenhagen.  For
a tour starting in Copenhagen, there is a 50\% chance that it'll go to
Amsterdam, a 30\% chance that it'll go to Berlin, and a 20\% chance to
remain in Copenhagen.  Assuming, for simplicity, that each tour will
be done in a day, what is the probability that the driver will spend
the night in each of the three cities?

(I will publish a sample solution on Monday, May 18.)

\item Absorbing Markov chains: Recognize an absorbing Markov chain.
Write the transition matrix in canonical form, solve for the
fundamental matrix and for the matrix of absorbing probabilities.
Find the mean exit times.

Typical problems: Homework 9, Question 5.


\item Two-player zero-sum games: Be able to check whether a game with
a given pay-off matrix is strictly determined.  If not, be able to
perform reduction-by-dominance if necessary, find the optimal mixed
strategies for both players, and determine the value of the game.

Typical problems: Homework 10, Question 3.


\end{enumerate}
\end{document}
