\magnification=\magstep1
\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Information on Exam 1}
\centerline{September 28, 2005}
\bigskip
\noindent
{\it General Information}
\bigskip
As announced in the course syllabus, the first midterm exam will
be given next week, either in class Friday, October 7.  The exam will
cover the material we have discussed since the beginning of 
the semester, up to and including the material on geometric
discrete random variables from class on September 28.
(This is the same as the material from Problem Sets 1 - 4 and 
Discussions 1 - 3).
In more detail, the topics to be covered are:
\bigskip
\item{1)}  Descriptive statistics such as the mean, standard
deviation, and frequency histograms for numerical data, the ``empirical rule''
\item{2)}  Discrete sample spaces and counting techniques
for sample points -- the $m\cdot n$ rule, permutations, 
binomial and multinomial coefficients 
\item{3)}  The ``Sample Point Method'' for probabilities
\item{4)}  Conditional probability and independence of events
\item{5)}  The Law of Total Probability and Bayes' Rule -- {\it know the
proofs of these} and how to apply them.  
\item{6)}  The ``Event Composition Method'' for probabilities
\item{7)}  Discrete random variables, probability distribution
functions, expected value, variance -- {\it know the proof of the 
equation $V(Y) = E(Y^2) - (E(Y))^2$ as well as how to apply it.}
\item{8)}  Binomial and geometric random variables -- {\it know the
proof of the formula $E(Y) = np$ for a binomial random variable
based on $n$ trials with success probability $p$}, and the 
formulas for variance of binomial random variables, plus 
expected value and variance of geometric random variables.
\bigskip
\noindent
{\it Other Groundrules}
\bigskip
You may prepare {\it one side of a $3\times 5$ inch index card}
with formulas and any other information you want to include, 
bring it to the exam and consult it at any time.  
I will provide copies of the tables for binomial random variables 
from the text.  Calculators allowed.
\bigskip
\noindent
{\it Review Session}
\bigskip
I will be happy to run a pre-exam review session if there is interest.
The evening of Tuesday, October 4 would be just about the only 
evening I can be here that week, though, because of off-campus
commitments.
\bigskip
\noindent
{\it Suggested Review Problems}
\bigskip
From the text:  1.22, 1.25, 1.33, 2.122, 2.123, 2.124, 2.127, 2.129, 2.131,
2.132, 2.137, 2.144, 2.145, 3.15, 3.33, 3.53, 3.56, 3.61, 3.144
\bigskip
\noindent
{\it Sample Exam}
\bigskip
\noindent
{\it This was the first midterm exam from the offering
of this course in Fall 2003.}
\bigskip
\noindent
I. A manufacturer of electronic components tests the lifetimes
of a certain type of battery and finds the following data:
$$123, 116, 122, 110, 125, 126, 111, 118, 117, 120$$
(lifetimes in hours).
How many of the sample points are within one standard deviation of 
the sample mean?  Is there reason to believe the lifetime
of this type of battery is not normally distributed from this small sample?
Explain.
\bigskip
\noindent
II.  In a regional spelling bee, the 10 finalists consist
of 5 girls and 5 boys.  Assume that all the finalists are equally 
proficient spellers and that the outcome of the contest is random.
What is the probability that 4 of the top 5 finishers will be female?
\bigskip
\noindent
III.  
\item{A)}  State and prove the Law of Total Probability.
\item{B)}  The Podunk City police department plans a crackdown
on speeders by placing radar traps at four different locations
$L_1,L_2,L_3,L_4$.  The probability that each of the 
traps is manned at any one time is $.4, .3, .2, .3$ respectively
(and the police really mean business -- if a trap is manned every speeder 
who passes it will get a ticket).  Speeders have probabilities 
of passing the four locations of $.2, .1, .5, .2$ respectively, 
and no one passes more than one.  What is the 
probability that a given speeder will actually receive a ticket?
\item{C)}  In the situation of part B, given 
that a speeder received a ticket, what
is the probability that he passed location $L_2$?
\bigskip
\noindent
IV.  Let $A,B$ be events for which $P(A) = .2$, $P(B) = .3$ and
$P(A\cap B) = .06$.  Are $\overline{A}$ and $\overline{B}$ independent
events?
\bigskip
\noindent
V.  An allergist knows that $30\%$ of all people are 
allergic to the pollen of burdock weed.
\item{A)}  Starting from some point in time, the first
five patients in sequence that 
the allergist sees are not allergic to burdock pollen.  
Given that, what is the probability that the first patient
the doctor sees who is allergic to burdock pollen will come
after the 20th patient seen?  Explain the assumptions you are making 
to derive your solution.
\item{B)}  What is the probability that from
4 to 7 (inclusive) of the next 20 patients she sees are allergic to 
burdock pollen?  Explain the assumptions you are making to derive 
your solution.
\bigskip
\noindent
{\it Extra Credit.} The Yankees won 10 of the 19 regular season games 
they played against the Red Sox in 2003.  Assuming that the Yankees
have a $10/19$ chance of winning each game they play against
the Sox, as of last Tuesday (i.e. before the first game), what 
was the probability that the Sox will clinch the ALCS at home?
Assume the games are independent events,  and recall that
the series starts with two games in New York, then moves to 
Boston for the next 3 (if needed),
then back to New York for the remainder of the series (if needed).
\bye

