Q2
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Probability
└── Homework
└── W9
└── Q2.tex
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\fancyhead[l]{Li Yifeng}
\fancyhead[c]{Homework \#9}
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\begin{document}
\section*{Question 2}
\noindent Suppose that women who live beyond the age of $80$ outnumber men in the same age group by three to one. How much information, in bits, is gained by learning that a person who lives beyond $80$ is male?
\bigskip
\begin{enumerate}[label={},leftmargin=0in]\item
\subsection*{Solution}
Let
\[
\begin{aligned}
\bP(W) &:= \text{probability of being a woman beyond the age of $80$}\\
\bP(M) &:= \text{probability of being a man beyond the age of $80$}
\end{aligned}
\]
Then we have
\[
\begin{aligned}
\bP(W) &= \frac{3}{4}\\
\bP(M) &= \frac{1}{4}
\end{aligned}
\]
So that
\[
I(X) = -log_2(\bP(M)) = 2 \text{bits}
\]
\subsection*{Answer}
\[\boxed{I(X) = 2 \text{bits}}\]
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\end{document}