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\hskip 1.5in \textbf{Midterm 1B--Stat 100--Spring 1997}     
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You may use calculators, but not books or notes. Each problem is 
worth 20 points.  Different parts of a problem have equal weight 
unless otherwise indicated. Do not spend too much time on any one problem. 
 Put a box around the final answer to a question. 
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1. (a) (15 pts.) Given is the sample data: 
$2,4,5, -5, -1.\ $   For this data, calculate  
the sample mean,  
sample standard deviation, and  
median.  
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(b) (5 pts.) 
  Suppose that data $x_1, x_2, \dots x_n$ has sample 
standard deviation equal to 6.  
What is the sample standard deviation  of the data 
$\ 3x_1 ,\  3x_2,\  \dots\  3x_n \ $ ? 
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2.  Suppose $A, B,$ and $C$ are events such that $P(A)=.5, P(B)=.8,
P(C)=.7,$ and $P(A|B)=.6$.  

(a) Find $P(AB)$. 

(b) Find $P(A \cup B)$. 

(c) Find $P(B|\overline{A})$.

(d)  Can $A$ and $C$ be mutually exclusive?  Why or why not?
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3.  Fifty high school students enter a
random drawing for two prizes.
  The first person drawn will win \$500, and the second person 
drawn will win \$100.  Suppose that 10 of the students are freshman,
15 are sophomores, and 20 are juniors.  
   
(a)  What is the probability that no freshman wins a prize? 

(b)  Suppose the first prize is won by a senior. Then what is the 
probability the second prize will be won by a junior? 
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4. Let X be the face shown by a loaded die and assume
the following probability distribution:
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\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline 
$n$ & 1 & 2 & 3 & 4 & 5 & 6  \\ \hline
$P([X=n])$ & 0.1 & 0.2 & 0.1 & 0.4 & 0.1 & 0.1
 \\ \hline
\end{tabular}
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\begin{enumerate}
\item
What is the probability of rolling an even number? 

\item
What is  the mean (expectation)   of X? 

\item 
What is the standard deviation of X? 

\item
Suppose you play the following game: on each roll, 
you receive 3 dollars, and you pay out the value shown 
in dollars.  How much do you expect to
win or lose if you play 15 times? 
\end{enumerate}

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5. Suppose a jar contains 5 balls labeled 1 through 5.
One after another, 20 people pick a ball at random,
announce the number on it and put it back into the jar.


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Find the probability of the following events. 
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\begin{enumerate}
\item
Ball 2 gets drawn no more than 4 times.
\item
Ball 2 gets drawn exactly 7 times.
\item
Ball 2 gets drawn at least 6 times.
\item 
Ball 2 gets drawn by the 7th and again by the  8th person.

\end{enumerate}
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