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\begin{document}
\mbox{}\vspace*{-1.5cm}

{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Mathematics 214: Introduction to Statistics }
\\
{\bf Spring Semester 1998}
\\
{\bf Exam 1}
\\
{\bf February 6, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}Examination
contains 15 multiple choice questions 6 points worth each.
You start with 10 points,  and the highest possible score is 100.
Fill in your answers on this cover sheet by placing an {\sf X} through one
letter for each problem.
Calculators, books, and notes are not allowed.
\wegdamit{
5 problems, each worth 20 points, on 7 sheets of paper
including the front cover and one extra sheet on the back.
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.
}
\\[1.7ex]
\hline
\end{tabular}
\vspace*{.7cm}

\mbox{}\hspace*{1.5cm}\begin{tabular}[t]{|p{.6cm}||p{.5cm}|p{.5cm}|p{.5cm}|p{.5c
m}|p{.5cm}||p{.8cm}|}\hline
 \ 1      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 2      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
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 \ 8      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular} \qquad\quad
\begin{tabular}[t]{|p{.6cm}||p{.5cm}|p{.5cm}|p{.5cm}|p{.5cm}|p{.5cm}||p{.8cm}|}\
hline
 \ \ 9      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 10      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
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% \ 16      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \end{tabular}
\\[1.3ex]
\mbox{}\hspace*{14.3cm} {\bf Total} \fbox{\rule[-.3cm]{0cm}{1cm}\qquad\ }

\vspace*{1.7cm}
{\bf Sign the pledge:}
\\[1ex]
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}

\newpage
{\bf 1)}
You roll a fair die twice in a row.
What is the probability that the sum of the upper face numbers is 8?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{1}{6}}$     &
{\bf b)} & ${\Ds \frac{{12 \choose 8}}{36}}$ &
{\bf c)} &  ${\Ds \frac{1}{12}}$  &
{\bf d)} & ${\Ds \frac{5}{36}}$ &
{\bf e)} & ${\Ds \frac{1}{{36 \choose 8}}}$
\end{tabular}
\vspace{1.4cm}

{\bf 2)}
The set of six letters $\{A,\,B,\,C,\,D,\,E,\,F\}$  is used to construct
4-letter words.
Repetitions of letters are allowed and the order counts, i.~e.,
the word $ABDD$ is different from the word $BDAD$.
Assume that the letters are chosen randomly, i.~e. that any letter chosen
from the set
$\{A,\,B,\,C,\,D,\,E,\,F\}$ is equally likely.
What is the probability that  a random word has all its 4 letters different?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{{6 \choose 4}}{6^4}}$     &
{\bf b)} & ${\Ds \frac{P^6_4}{6^4}}$ &
{\bf c)} &  ${\Ds \frac{1}{6^4}}$  &
{\bf d)} & ${\Ds \frac{4!}{6!}}$ &
{\bf e)} & ${\Ds \frac{{6 \choose 4}}{6!}}$
\end{tabular}
\vspace{1.4cm}

{\bf 3)}
There is a well-shuffled standard card deck of 52 cards.
What is the probability that a poker hand of five cards dealt randomly from
the deck
contains exactly 3 aces.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{{4 \choose 3}\cdot{48 \choose 2}}{{52 \choose 5}}}$
&
{\bf b)} & ${\Ds \frac{5\cdot4\cdot3\cdot 47^2}{52^5}}$ &
{\bf c)} &  ${\Ds \frac{{4 \choose 3}\cdot48^3}{{52 \choose 5}}}$  &
{\bf d)} & ${\Ds \frac{{4 \choose 3}+{48 \choose 2}}{{52 \choose 5}}}$ &
{\bf e)} & ${\Ds \frac{4^3\cdot47^2}{52^5}}$
\end{tabular}
\vspace{1.4cm}

\wegdamit{
{\bf 4)}
There is a well-shuffled standard card deck of 52 cards.
What is the probability that a poker hand of five cards dealt randomly from
the deck
contains no aces.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{5\cdot48^4}{52^5}}$     &
{\bf b)} & ${\Ds \frac{P^{48}_5}{52^5}}$ &
{\bf c)} &  ${\Ds \frac{{48 \choose 5}}{{52 \choose 5}}}$  &
{\bf d)} & ${\Ds \frac{{48 \choose 4}{5 \choose 4}}{{52 \choose 5}}}$ &
{\bf e)} & ${\Ds \frac{48^5}{52^5}}$
\end{tabular}
\vspace{1.4cm}
}

{\bf 4)}
Assume $A$ and $B$ are events, i.~e. subsets of a sample space $S$.
Let $P$ be a probability defined on $S$.
Assume that $P(A)=0.45$, $P(B)=0.15$, and $P(A\cap B)=0.03$.
Calculate $P(A\,|\,B)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{1}{2}}$     &
{\bf b)} & ${\Ds 0.45}$ &
{\bf c)} &  ${\Ds \frac{1}{3}}$  &
{\bf d)} & ${\Ds 0.03}$ &
{\bf e)} &  ${\Ds \frac{1}{5}}$
\end{tabular}
\vspace{1.4cm}

{\bf 5)}
In a certain company 60\% of the employees are female and the remaining
40\% are,
of course, male.
30\% of the female employees  and 40\% of the male employees are younger than
35 years of age.
What fraction of all the employees of that company are younger than 35
years of age?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & $0.32$     &
{\bf b)} & $0.5$ &
{\bf c)} &  $0.36$  &
{\bf d)} & $0.7$ &
{\bf e)} &  $0.34$
\end{tabular}

\newpage
{\bf 6)}
There is a box of nails.
Assume that $\frac{2}{3}$ of these nails were produced by a machine (call
it A) that
is known to produce defective nails with probability 0.01, and the
remaining $\frac{1}{3}$
of the nails were produced by a machine (call it B) that is known to
produce defective nails
with probability 0.03.
If a nail is chosen randomly from the box and it is found to be defective,
what is the
probability it was produced by machine A?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 1}$     &
{\bf b)} & ${\Ds 0.6}$ &
{\bf c)} &  ${\Ds 0.4}$  &
{\bf d)} & ${\Ds 0.5}$ &
{\bf e)} & ${\Ds 0.2}$
\end{tabular}
\vspace{1.4cm}

{\bf 7)}
At a certain institute 55\% of the members have cookies in the afternoon,
40\% of the members have tea in the afternoon, and 20 \% of the members
have neither.
What fraction of the members like both cookies and tea?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & $40 \%$     &
{\bf b)} & $15 \%$ &
{\bf c)} &  $20 \%$  &
{\bf d)} & $95 \%$ &
{\bf e)} & $80 \%$
\end{tabular}
\vspace{1.4cm}

{\bf 8)}
A company produces light bulbs and ships boxes consisting of 40 bulbs.
Before shipping, a random sample of 6 bulbs is taken from the box and inspected
--- if any are defective,  the box is not shipped.
Assuming the box contains 8 defective bulbs, what is the probability that
the box won't be
shipped, i.~e., that the random sample of 6 bulbs contains at least one
defective.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds\frac{32^6-8^6}{40^6}}$     &
{\bf b)} & ${\Ds\frac{{8\choose 6}}{{40 \choose 6}}}$ &
{\bf c)} &  ${\Ds\frac{{32\choose 6}}{{40\choose 6}}}$  &
{\bf d)} & ${\Ds1-\frac{{32\choose 6}}{{40\choose 6}}}$ &
{\bf e)} & ${\Ds\frac{{8\choose 1}\cdot{32\choose 8}}{{40\choose 8}}}$
\end{tabular}
\vspace{1.4cm}

{\bf 9)}
39 \% of a  given population has type A blood.
A specific blood typing test predicts type A blood for 37\% of the population.
If a person from the population has type A blood, then the test will
predict with probability 0.91
that the person's blood is type A.
What is the probability that a certain person actually has type A blood
given that the test predicts
type A blood for that person?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & $0.91\cdot0.38$   &
{\bf b)} & ${\Ds\frac{0.91\cdot0.39}{0.37}}$ &
{\bf c)} &  ${\Ds\frac{0.91\cdot0.37}{0.39}}$  &
{\bf d)} & $1$ &
{\bf e)} & $0.91\cdot0.37$
\end{tabular}
\vspace{1.4cm}

{\bf 10)}
Assume that $A$ and $B$ are events of a sample space $S$.
The probabilities $P(B)=0.4$, $P(A\,|\,B)=0.3$, and $P(A\,|\,\bar{B})=0.6$
are known.
What is $P(A)$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & $0.48$     &
{\bf b)} & $0.12$ &
{\bf c)} &  $0.24$  &
{\bf d)} & $0.7$ &
{\bf e)} & $0.9$
\end{tabular}
%\vspace{1.4cm}

\newpage
{\bf 11)}
Assume $A$ and $B$ are disjoint events, i.~e., $A$ and $B$ are subsets of a
sample space $S$ and
the intersection $AB=A\cap B=\emptyset$.
Let $P$ be a probability defined for the sample space $S$.
If further $A$ and $B$ are independent events, then it is always true that
$P(A)\cdot P(B)=$

\begin{tabular}{cp{2.8cm}cp{2cm}cp{2cm}cp{2cm}cp{3.2cm}}
{\bf a)} & $P(A)+P(B)$     &
{\bf b)} & ${\Ds\frac{1}{2}}$ &
{\bf c)} &  $1$  &
{\bf d)} & $0$ &
{\bf e)} & $1-P(A)-P(B)$
\end{tabular}
\vspace{1.4cm}

{\bf 12)}
Let $X$ be a discrete random variable which takes the values $-1,\,2$, and $4$.
Its  probability function $p$ is given as $p(-1)=0.4,\; p(2)=0.3$, and
$p(4)=0.3$.
What is $E(X)$, i.~e., what is the expected value of $X$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & $2$     &
{\bf b)} & $1$ &
{\bf c)} &  $5$  &
{\bf d)} & $\sqrt{1.4}$ &
{\bf e)} & $1.4$
\end{tabular}
\vspace{1.4cm}

{\bf 13)}
How many ways can 3 different books be given to 7 persons, if each person is
allowed to receive several books?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & $7^3$     &
{\bf b)} & $3^7$ &
{\bf c)} &  ${\Ds{7\choose 3}}$  &
{\bf d)} & ${\Ds P^7_3}$ &
{\bf e)} & ${\Ds3\cdot 7}$
\end{tabular}
\vspace{1.4cm}

{\bf 14)}
Let $X$ be a discrete random variable which takes the values $-1,\,2,\,3$,
and $4$.
The probability function of $X$ is given as
$p(-1)=0.1,\;p(2)=0.2,\;p(3)=0.1$, and $p(4)=0.6$.
What is the standard deviation of $X$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} &  ${\Ds 2.4}$   &
{\bf b)} & ${\Ds 2}$ &
{\bf c)} &  ${\Ds\sqrt{2.4}}$  &
{\bf d)} & ${\Ds 1.2}$ &
{\bf e)} & ${\Ds \sqrt{3}}$
\end{tabular}
\vspace{1.4cm}

{\bf 15)}
Of the people entering a blood bank to donate blood, 30\% have type O$^+$
blood.
For the next three people entering, let $X$ denote the number with O$^+$ blood.
What is $P(X=2)$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} &  ${\Ds 3\cdot(0.3)^2\cdot0.7}$   &
{\bf b)} & ${\Ds 3\cdot(0.7)^2\cdot0.3}$ &
{\bf c)} &  ${\Ds (0.3)^2}$  &
{\bf d)} & ${\Ds  (0.7)\cdot0.3}$ &
{\bf e)} & ${\Ds  (0.3)^2\cdot0.7}$
\end{tabular}
\vspace{1.4cm}

\wegdamit{
Five employees of a firm are ranked from 1 to 5 based on their ability to
program a computer.
Three of these employees are selected to fill three identical programming jobs.
If all possible choices of three (out of the five) are equally likely, what
is the
probability that the highest-ranked employee among those selected has rank 2?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds\frac{3}{10}}$     &
{\bf b)} & ${\Ds\frac{1}{5}}$ &
{\bf c)} &  ${\Ds\frac{P^5_2}{5!}}$  &
{\bf d)} & ${\Ds\frac{P^5_3}{5^3}}$ &
{\bf e)} & ${\Ds\frac{2}{5^3}}$
\end{tabular}
\vspace{1.4cm}
}
\end{document}
