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{\bf
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MA 162 & \hspace{2.8in} & Russell Brown \\
Exam 3 & \hspace{2.8in} & 23 April, 1993
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Answer all of the following questions.
Use the backs of the question papers for scratch paper.
Additional sheets are available if necessary.  No books
or notes may be used.  When answering these questions,
please be sure to 1) check answers when possible, 2)
clearly indicate your answer and the reasoning used to
arrive at that answer {\bf (unsupported answers will receive
NO credit)}, and 3) label all variables and equations.

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\begin{center}
\begin{tabular}{|r|c|r|}
\hline
Question & Score & Total \\
\hline \hline
1 && 20 \\ & & \\
\hline
2 && 20 \\ & & \\
\hline
3 && 20 \\ & & \\
\hline
4 && 20 \\ & & \\
\hline
5 && 20 \\ & & \\
\hline \hline
Total && 100 \\
\hline  
\end{tabular}
\end{center}

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\noindent
Name \underline{$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $}
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\noindent
Section \underline{$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $}
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\begin{enumerate}

\item
a)  Compute:
\vspace{.2in}

(i) The number of three letter words which can
be formed using the letters in `BROWN' if we do not
allow repeated letters.
\vspace{1.2in}

ii) The number of three letter words which can be formed using
`BROWN' if we do allow repeated letters.
\vspace{1.2in}

iii)  The number of four letters words which can be formed
by using the four letters in `TOOT'.
\vspace{1.2in}

b)  Using a tree, list all of the ways that Boston
and Los Angeles can play a best of three series.
How many different possibilities are there?
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\newpage

\item
a)  A coin is flipped 5 times.  What is the probability of obtaining
at least one head?
\vspace{3.5in}

b) Suppose that we roll two dice.  What is the 
probability that the first die is a 3 or the second die is a 5?
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\item
Suppose that $A$ and $B$ are events with $ P(A) = 0.5 , ~~
P(B) = 0.6 $ and $ P ( A \cup B ) = 0.7 $.
\vspace{.2in}

a)  What is $ P ( A \cap B ) $?
\vspace{2.7in}

b)  Complete the definition:  The events $A$ 
and $B$ are independent if \underline{$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$}.
\vspace{2.7in}

c)  Using the definition you gave in b) determine if
the events $A$ and $B$ are independent.
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\newpage

\item
Suppose that a carnival game involves attempting five free
throws.  Assume that the probability of making one free throw
is $ 0.4 $.
\vspace{.2in}

a)  What is the probability of making exactly five throws?
\vspace{3.5in}

b)  Suppose that a prize of \$20 is given to players making all five
free throws and a prize of \$5 is given to those
who make exactly four free throws.  What should
they charge to play this game 
in order to break even?
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\newpage

\item
Disease $X$ is found in 20\% of the patients at a clinic.
There is a test for disease $X$ which is known to give a positive
result (indicating the presence of disease $X$) in
80\% of the ill patients and is also positive in 25\% of the 
patients who do not have disease $X$.
\vspace{.2in}

a) Let $I$ denote the event of patients with disease $X$ and 
let $+$ and $-$ denote the events of patients who test positive and 
negative.  Draw a tree which 
summarizes the above information and then compute $ P ( I \cap + ) $
and $ P ( I ' \cap - ) $.
\vspace{3.3in}

b) Suppose that a patient tests positive.  What is 
the probability he or she has disease $X$?
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\end{enumerate}
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