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MA162
\hfill
Review \quiznumber 

\end{flushleft}
The following topics and skills will be examined in the third midterm test.
\begin{itemize}
\item Trees, the multiplication principle, Venn diagrams and the
inclusion-exclusion principle and counting problems.
\item  Permutations, combinations and distinguishable permutations.
You will need to know the formulas and be able to recognize which
formula is needed in a particular problem.
\item Computation of probability using counting techniques.  Use of
the complement rule and the formula for the probability of the union
of two events.
\item Definitions of conditional probability and independent events. 
%\item Expected value of a random variable.
\item Bayes formula.  The use of a tree in organizing the information
for Bayes theorem.
%\item Binomial probability.
\end{itemize}

Below are some sample questions to help you prepare for the exam.
Be sure that you understand all of the topics from the above list, not
just the topics covered by the questions below!
In your answers on the exam,
please be sure to 1) Check answers when possible. 2) Clearly indicate
your answer and the reasoning used to arrive at that answer. 3)  Label
all variables and equations. 



\begin{enumerate}
\item Is it always true that $B\cap ( A\cup C)'= (A' \cup B ) \cap (
A' \cup C)$? Draw two Venn diagrams to justify your answer.


\item Five thousand people answered a survey about vacations spots. Of
these, 1655 liked Florida, 1155 liked California and 955 liked Hawaii.
Both California and Florida were preferred by  105, 55 liked both
California and Hawaii and 80 liked both Hawaii and Florida.  There
were 30 people who liked all three places.  
\begin{enumerate}
\item How many liked just Hawaii?
\item How many liked Florida and California, but not Hawaii?
\item How many did not like any of the three?
\end{enumerate}

\item
\begin{enumerate}
\item If a lottery has 40 numbers and you choose six of these for each
play, how many possible tickets are there?  Assume that the order
does not matter.
\item How many distinguishable permutations are there  of the letters
in `COMMITTEE'?
\item Given that the number of combinations of  5 distinct objects
taken two at a time  is
$5!/(3!2!)$, explain why the number of ways of arranging the 5 letters
in `ALOOF' 
is 5!/2.


\item Seventy people belong to the math club.  How many different ways
can we choose a speaker committee of 4 people and a publicity
committee of 6.  Assume that no one is expected to serve on more than
one committee.
\end{enumerate}

\item A coin is flipped. If the coin is heads, you choose a number
between 1 and 3.  If the coin is tails, you choose a number between 1
and 4. Draw a tree to summarize this probability experiment and answer
the following questions.

\begin{enumerate}
\item List the elements in the sample space for this game. Are all the
outcomes equally likely?
\item What is the probability of obtaining a 1?
\item What is the probability of obtaining a 4?
\item What is the probability of obtaining an even number?
\end{enumerate}

\item If a coin is flipped 5 times, compute the probability of
\begin{enumerate}
\item obtaining no heads.
\item obtaining at least one head.
\item obtaining exactly one head.
\end{enumerate}


\item A survey of 100 students gives the following information.  
\begin{itemize}
\item 55 took calculus \item 40 took finite mathematics \item 15 did
not take calculus and did not take finite mathematics
\end{itemize}
\begin{enumerate}
\item Find the number of students who took calculus and finite mathematics.
\item Given that a student took calculus, find the probability that
the student also took finite mathematics.
\item Find the probability that a student took finite mathematics or calculus.
\item According to the above information, are the events of taking
finite mathematics and calculus independent?
\end{enumerate}

\item Suppose that you draw a number from 1,2,3, 4, 5.  If you draw an
odd number, you receive that amount in dollars. If you draw an even number,
you must pay that amount.  What is the expected value of playing this game?


\item Suppose that 30\%  of patients with a sore throat have a
bacterial infection.  Suppose that a throat culture indicates bacterial
infection in 
10\% of  patients without bacterial infections and does not indicate
bacterial infection in 
15\% of those   who have a bacterial infection.
\begin{enumerate}
\item Draw a tree to summarize the above information.
\item What is the probability that a patient with a sore throat will
have a positive culture?
\item Suppose that a patient has a positive culture.  What is the
probability that   the patient has a bacterial infection?
\end{enumerate}



\comment{
\item Find the expected gain of playing the following lottery.
\begin{center}
\begin{tabular}{l|lllllll}
Payoff & \$500 & \$250 & \$100 & \$50 & \$ 10 & \$5 & \$1 \\
Number of tickets& 5 & 10 & 15&25& 50& 100 &200\\
\end{tabular}
\end{center}
There are a total of 1000 tickets.


\item \begin{enumerate}
\item a friend bets you that if you roll a die 12 times, you will not
get exactly 2 fives.  what is the probability of obtaining exactly two
fives?
\item what is the probability that you will obtain at least 2 five's?
      \end{enumerate}



}
\end{enumerate}

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