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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Computing Probabilities -- Example Problems}
\centerline{September 17, 2009}
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\noindent
{\it Background}
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Last time we introduced the conditional probability:
$$
P(A|B) = {P(A\cap B)\over P(B)}
$$
and several other useful facts:
\item{1)}  The "multiplicative rule":  $P(A\cap B) = P(A|B)P(B)$
\item{2)}  The "additive rule":  $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
\item{3)}  The "complement rule":  $P(\overline{A}) = 1 - P(A)$.
Today we want to practice using these on the following examples.
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\noindent
{\it Questions}
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\item{A)}  (Hypothetical!)  Polls show that $70\%$ of Democrats and 
$20\%$ of Republicans favor the current health care reform proposals.
The population consist of $55\%$ Democrats and $45\%$ Republicans.
A single person is chosen randomly.  What is the probability that
she is a Democrat, given that she favors the health care reform 
proposals?
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\item{B)}  As items come to the end of a production line, an inspector
chooses which items are to go through a complete inspection.  $10\%$
of all the items are defective.  $60\%$ of the items that are defective
go through a complete inspection and $20\%$ of the items that are not 
defective go through a complete inspection.  What is the probability
that an item is defective, given that that it is chosen for a complete 
inspection. 
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\item{C)}  (From the Society of Actuaries "P Exam" Study Materials)  Of 937 men
who died in 1999, 210 died of causes related to heart disease.  312 of the
937 men had at least one parent who had heart disease.  Of those 312, 102 died of
causes related to heart disease.  If a man is randomly selected from the 
group, what is the probability that he died of heart disease, given that
neither parent had heart disease?  
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\item{D)}  An accident victim will die unless he receives type $A+$ blood
before 10 minutes elapse, and he will be saved if he does get the transfusion.
Potential blood donors and a reusable blood typing kit are available. 
The donors do not know their blood types, though.  It 
takes 2 minutes to determine each one's type and only one can be 
processed ata time.  Only
$40\%$ of the potential donors have type $A+$ blood.  What is the probability
that the victim will be saved?
\bye
