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\begin{document}





\author{Brian C. Hall}

\title{Math 214--Test 1}

\date{February 23, 2001}

\maketitle







\textbf{Instructions}. There are 9 problems worth 11 points each, with 1 free

points, for a total of 100 points possible. Answer all the questions.

\textit{Show your work on all problems.} Calculators are permitted. Notes and

books are not permitted. You have one hour for the test.



This test is subject to the Notre Dame honor code.\bigskip



\textbf{Name}:\bigskip



\begin{enumerate}

\item  a) If $A$ and $B$ are any two events, what is the formula for the

probability that both $A$ and $B$ occur?



b) How do you tell mathematically whether or not $A$ and $B$ are independent?

(I am looking for a formula that characterizes when $A$ and $B$ are independent.)\pagebreak 



\item  A newspaper keeps statistics for one year on the free-throws of a

particular player. On occasions when the player shoots two free-throws the

paper found: 90\% of the time the player makes the first shot; 85\% of the

time the player makes the second shot; 3\% of the time the player misses both shots.



a) Are the first shot and the second shot independent? Why or why not?



b) What percentage of the time does the player make both shots?\pagebreak 



\item  Suppose $X$ is a random variable with the following probability

distribution.

\[%

\begin{array}

[c]{ccc}%

\text{possible value }x &  & \text{probability }p(x)\\

0 &  & 0.10\\

1 &  & 0.20\\

2 &  & 0.20\\

3 &  & 0.40\\

4 &  & 0.10

\end{array}

\]

Compute the mean, variance, and standard deviation of $X.$\pagebreak 



\item  A jar contains 6 red balls and 4 white balls. If you select 3 balls at

random without replacement, what is the probability that all 3 balls will be

of the same color?\pagebreak 



\item  What is the minimum number of times you should roll a fair die so that

the probability of getting at least one six is at least $\frac{1}{2}$?\pagebreak 



\item  If you roll a fair die 10 times, what is the probability of getting

exactly 4 sixes? What is the probability of getting at least 3 sixes?\pagebreak 



\item  A computer sends 1,000,000,000 bytes of information across a network.

If the probability of an error on any one byte is 1/400,000,000 and the bytes

are independent, what is the probability that there will be exactly 4 errors?\pagebreak 



\item  Parts produced by a given machine are inspected by two different

inspectors. The first inspector catches 93\% of the defective parts. Of the

defective parts missed by the first inspector, the second inspector catches

60\%. What is the probability that a defective part will be missed by both inspectors?\pagebreak 



\item  The number $X$ of cancer cases per year in a given county is a Poisson

random variable with mean $1.6.$



a) What is the standard deviation of $X$?



b) What is the smallest integer $n$ such that $P(X<n)$ is at least 94\%?

\end{enumerate}

\end{document}