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\centerline{Mathematics 375 -- Probability Theory}
\centerline{Midterm Exam 1 -- October 7, 2011}
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{\it Directions:}  Do all work in the blue exam booklet.  There
are 100 possible points and 10 Extra Credit points.  Note:  If an 
answer can be expressed in terms of factorials, binomial coefficients,
multinomial coefficients etc., please {\it leave it in that form} 
rather than evaluating it further.  
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I.  (15)  To test the effectiveness of a seal designed to keep a tire patch
airtight, a needle was inserted into the tire and air pressure
was increased until leakage was observed.  The pressures (in lb per square
inch) where leakage first occurred on 10 trials were:
$$8.1, 8.5, 8.3, 8.8, 8.2, 7.8, 8.9, 7.5, 8.3, 8.0$$
How many of the data points are within 1 standard
deviation of the mean?  How well does this fit the ``empirical rule?''
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II.  Students on a boat send signals back to the shore by stringing
(exactly) 9 colored flags along a vertical rope (without overlapping).
The signal is determined solely by the relative positions of the flags.  
\item{A)} (5) How many different signals can they send if
they have flags of 9 different colors?
\item{B)} (10)  Now suppose they have 4 red flags and 5 green flags. If a random 
arrangement of flags is constructed, what is the probability that
all the green flags appear {\it consecutively}?
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III.   The first three parts of this question deal with this
situation:  Let $A_1,A_2,A_3$ be events in a sample space $S$.
Let $P(A_1) = .4$, $P(A_2) = .2$, $P(A_3) = .4$, and assume
$P(A_i \cap A_j) = \emptyset$ if $i \ne j$.  
Finally, let $B$ be another event with $P(B|A_1) = .1$,
$P(B|A_2) = .2$ and $P(B|A_3) = .05$.  
\item{A)} (10) What is $P(B)$?
\item{B)} (5) Are $B$ and $A_2$ independent events?  Why or why not?
\item{C)} (10) Find $P(A_1 \cup A_2 | B)$.
\item{D)} (15) Now, working in general, state and prove Bayes' Rule.  
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IV.  
\item{A)}  (20)  A jar contains 7 red and 11 white balls.  Draw {\it two
balls} at random, {\it replacing the first ball after it is drawn}.  
Let $X$ be the number of balls drawn that are red.  Thinking of $X$ as a 
discrete random variable, find its probability function, and compute 
its expected value and variance.
\item{B)}  (10)  Now consider drawing two balls {\it without replacing 
the first ball after it is drawn}, and let $Z$ be the number 
of red balls drawn.  Give the probability function for this $Z$.  
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{\it Extra Credit}  (10)
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The probability that a randomly chosen male has a circulation problem
is $.25$.  Males who have a circulation problem are twice as 
likely to be smokers as those who do not have a circulation problem.
What is the probability that a male has a circulation problem, given
that he is a smoker?   
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