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{\bf Worksheet \#23; date: 11/26/2018}
{\bf MATH 55 Discrete Mathematics}
\begin{enumerate}
\item {\em (Rosen 7.3.5)} Suppose that $8\%$ of all bicycle racers use steroids, that a bicyclist who uses steroids tests positive for steroids $96\%$ of the time, and that a bicyclist who does not use steroids tests positive for steroids $9\%$ of the time. What is the probability that a randomly selected bicyclist who tests positive for steroids actually uses steroids?
\item {\em (Rosen 7.3.10)} Suppose that $4\%$ of the patients tested in a clinic are infected with avian influenza. Furthermore, suppose that when a blood test for avian influenza is given, $97\%$ of the patients infected with avian influenza test positive and that $2\%$ of the patients not infected with avian influenza test positive. What is the probability that:
\begin{enumerate}
\item a patient testing positive for avian influenza with this test is infected with it?
\item a patient testing positive for avian influenza with this test is not infected with it?
\item a patient testing negative for avian influenza with this test is infected with it?
\item a patient testing negative for avian influenza with this test is not infected with it?
\end{enumerate}
\item The prevalence of breast cancer among $30$ year old women is $1\%$. Mammogram as a screening test shows positive $80\%$ of the time on women with breast cancer, but $10\%$ on women without breast cancer. If a $30$ year old woman is tested positive, how worried should she be? Suppose we want $90\%$ of the women who are tested positive to actually have cancer, how much false positive should we tolerate?
\item A satellite transmits one of three messages using a bitstring as its code. Suppose the three messages are encoded as $00$, $01$ and $10$, and each bit has a $1/3$ chance of being corrupted. Suppose also all three codes are used equally as frequent. What is the probability that:
\begin{enumerate}
\item the sent message is $00$ given that the received message is $00$?
\item the sent message is $01$ given that the received message is $01$?
\item the sent message is $00$ given that the received message is $11$?
\end{enumerate}
\end{enumerate}
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