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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Binomial and Geometric Random Variables}
\centerline{October 1, 2009}
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{\it Background}
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Last time we introduced two types of discrete random variables -- 
{\it binomial} and {\it geometric} variables.  Both deal with
the situation where independent trials of the same process
are repeated, and the probability of success in any one trial is
$p$.
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\item{1)}  The prototypical 
{\it binomial} random variable is the number of ``successes''. 
The probability mass function for a binomial $Y$ looks like
$$p_Y(y) ={n \choose y} p^y q^{n-y},$$
where $q = 1 - p$ is the complemetary (``failure'') probability.
\item{2)}  The prototypical {\it geometric} random variable
is the number of the trial on which the first success is 
observed.  The probability mass function for a geometric $Z$ looks like
$$p_Z(z) = q^{z-1}p.$$
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\noindent
Today, we will deal with both types of situations, and practice
telling them apart.
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{\it Example Questions}
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A)  An oil prospector drills a succession of wells in a given
area.  The probability that she is successful on a given trial
is $p = .2$, and the trials are independent.
She has financing to drill 10 wells in all.
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\item{1)}  On average, how many wells would she expect to have
to drill before finding a productive well?
\item{2)}  On average, how many productive wells would she 
expect to find among the 10?
\item{3)}  
What is the probability that she will find at least 2 productive wells?
\item{4)}  What is the probability that she will fail to find a 
productive well?
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\noindent
B)  In responding to a sensitive question such as ``Have you 
ever used marijuana?'' on a survey, many people prefer not to answer
``yes'' even if that is true.  Suppose $80\%$ of the population will 
truthfully answer ``no'', and of the $20\%$ who should truthfully
answer ``yes'', $70\%$ will lie.
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\item{1)}  If 100 people are selected randomly and asked this
question, what is the expected number of ``yes'' responses?
\item{2)}  What is the expected number of people you would
need to question before obtaining a ``yes'' response?
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\noindent
C)  Of the volunteers donating blood at the Worcester Red Cross,
$80\%$ have the Rh ($+$) factor (i.e. one of the blood types $A+, B+, O+, AB+$).
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\item{1)}  If 5 volunteers are randomly selected, what is the 
probability that least one does not have the Rh factor?
\item{2)}  What is the probability that 6 volunteers will
be tested before we find one that does have the Rh factor?
\item{3)}  Say you want to be ``90\% sure'' that you 
have at least 5 donors who do have the Rh factor.  What
is the smallest number of volunteers you would need to 
select in order for the probability that at least 5 have
the Rh factor to be at least .9?  (Note: You can use Maple
or a calculator here as you prefer.  The binomial tables in
our text don't cover all the cases you need.)
\bye