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

\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Mathematics 214: Introduction to Statistics }
\\
{\bf Review for Exam 1, February 1999}
\end{center}

{\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}

\junk{
{\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{.8cm}

{\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{.8cm}

{\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{.8cm}

{\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{.8cm}

{\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{.8cm}

{\bf 16}
A certain manufacturer advertises batteries that will run for an average of 150 minutes, with
a standard deviation of 10 minutes.
Find the smallest interval, which contains at least 75\% of the performance periods for 
batteries of this type?  
%Which of the following intervals contains at least 75\% of the performance periods for batteries
%of this type?  

\begin{tabular}{cp{3cm}cp{3cm}cp{3cm}cp{3cm}}
{\bf a)} & ${\Ds 50\leq Y\leq 250}$     &
{\bf b)} & ${\Ds 140\leq Y\leq 160}$ &
{\bf c)} &  ${\Ds 110\leq Y\leq 190}$  & 
{\bf d)} & ${\Ds 130\leq Y \leq 170}$ \\[3ex] 
{\bf e)} & ${\Ds 145\leq Y\leq 155}$
\end{tabular}

\newpage
\begin{center}
\mbox{}\hspace*{1.5cm}\begin{tabular}[t]{|p{.6cm}||p{.5cm}|p{.5cm}|p{.5cm}|p{.5cm}|p{.5cm}|}\hline
 \ 1      &\  a &\ b &\ c &\ X &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 2      &\  a &\ X &\ c &\ d &\ e  \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 3      &\  X &\ b &\ c &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 4\rule[-.5cm]{0cm}{1.2cm}     &\  a &\ b &\ c &\ d &\ X     \\ \hline
 \ 5\rule[-.5cm]{0cm}{1.2cm}      &\  a &\ b &\ c &\ d &\ X     \\ \hline
 \ 6      &\  a &\ b &\ X &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 7      &\  a &\ X &\ c &\ d &\ e   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 8      &\  a &\ b &\ c &\ X &\ 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}|}\hline
 \ \ 9      &\  a &\ X &\ c &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 10      &\  X &\ b &\ c &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 11      &\  a &\ b &\ c &\ X &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 12      &\  a\rule[-.5cm]{0cm}{1.2cm}     &\ b &\ c &\ d &\ X    \\ \hline
 \ 13      &\  X &\ b &\ c &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 14      &\  a &\ b &\ X &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 15      &\  X &\ b &\ c &\ d &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 16      &\  a &\ b &\ c &\ X &\ e    \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \end{tabular}
\end{center}

\end{document}