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\begin{document}
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\centerline{Math 36 --- Weekly Homework}
\centerline{Assigned: 10/15}
\centerline{Due: 10/22}
\begin{questions}
\question
Should we expect vote totals in voting precincts to follow Benford's Law? Why or why not?
\vspace{2 in}
\question
Suppose you are on a Monty Hall gameshow with with 100 doors. After you pick a single door, the host opens 70 doors.
\begin{enumerate}[label = (\alph*)]
\item What is the probability that the door you chose has the prize?
\item What is the probability that any one of the unpicked, unopened doors has the prize?
\item Find the answer to (b) for a more general game with $n$ total doors and $m$ opened doors.
\end{enumerate}
\vspace{4 in}
\question
In this problem, we will use our new knowledge about Markov chains to analyze the Birth-death process. Suppose we have a population with 5 individuals, one type A with fitness 2 and the other four being type B with fitness 1. We will make this a Markov chain where state $i$ means $i$ individuals of type A. (You are free to use a computer to do the matrix calculations, particularly the matrix inverse.)
\begin{enumerate}[label = (\alph*)]
\item Draw this Markov chain. Include all the transition probabilities.
\item Which states are absorbing? Which states are transitive?
\item Give the transition matrix $P$ in \textbf{canonical form}. Be sure to label the rows and columns so it is clear which state is being represented.
\item Give the matrices $Q$, $R$, $N = (I-Q)^{-1}$, and $B = NR$.
\item What is the probability that type A takes over the population?
\item What is the expected value for the time it will take for A to either fixate or go extinct?
\end{enumerate}
\end{questions}
\end{document}
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