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\large{\textbf{Math 4600/6600}}
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\section{Problems about Markov Chains}
Here are some problems about Markov chains.
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\item \textbf{Jesse Presley} The singer Elvis Presley had a twin brother (Jesse Garon Presley\footnote{Sadly, Jesse Presley died at birth, which may explain why not many people know about him.}). At the time, 15 births of every 1000 were births of twins, while 3 of every 1000 births were births of identical twins. What is the probability that Elvis was an identical twin? Please write several sentences to explain your reasoning.
\item \textbf{Are rats random?} According to the CDC, ``Compared to nonsmokers, men who smoke are about 23 times more likely to develop lung cancer and women who smoke are about 13 times more likely.'' If you learn that a woman has been diagnosed with lung cancer, and you know nothing else about her, what is the probability that she is a smoker? Note that you will have to do some research to solve this problem. Please determine what additional information you need, find it, and cite a source. Again, write several sentences to explain your reasoning.
\item \textbf{The distracted driver} You've already analyzed the data in ``CoinFlipData.csv'' on the course webpage to arrive at a Bayesian estimate of the probability that the coin is biased. Now plot your confidence level $P(\text{biased})$ as a function of the number of flips that you have seen. Turn in the graph.
\item \textbf{The Bayesian Filter} Suppose that we are given a sequence of coin flips from a coin which has probability $p$ of returning ``heads'' and probability $1-p$ of returning tails. We will consider 11 different options for modeling the coin: $p = 0, \frac{1}{10}, \frac{2}{10}, \dots, 1$. We will call these events $A_0, \dots, A_{10}$ and assume that they are the only possibilities for the coin (that is, that they are a partition of the probability space).
At the start of the trial, we assign initial probabilities $P(A_i) = \frac{1}{11}$. After each flip $B$, we use Bayes' theorem to update all 11 of these probabilities using the formula
\begin{equation*}
P(A_i|B) = \frac{P(A_i) P(B|A_i)}{\sum_{j=0}^{10} P(A_j) P(B|A_j)}
\end{equation*}
The file ``BiasedCoinData.csv'' on the course webpage contains a large collection of data from such a coin.
Use a computer to analyze the data and plot the probabilities $P(A_i)$ (as a function) of $i$ after 0 flips, 10 flips, 100 flips, 1,000 flips, 10,000 flips, and all 100,000 flips. Be sure to include a printout and explanation of your code or your spreadsheet.
\item \textbf{The (over?)confident Bayesian} Suppose that you are completely certain that a belief you hold is true: your prior belief $P(A) = 1$. You observe an event $B$ which is extremely unlikely, given your theory of the world: $P(B|A) = 0.0001$. Compute your updated level of belief in $A$. Does $P(B|\not A)$ matter to you? Why or why not? Can any piece of evidence ever change your mind?
\item \textbf{An unlikely story} While visiting Loch Ness on vacation, you meet Mr.\ MacWaffle, a local who claims to have seen Nessie out for a swim last Tuesday, an event to which you assign the low prior probability of $1/1000$. In your mind, there are two options:
\begin{itemize}
\item $A$: MacWaffle is completely honest and would claim to see Nessie if and only if he saw the monster.
\item $A^c$: MacWaffle is desperate to drum up business for his soft-serve ice cream shop and would have reported a monster sighting no matter whether or not he saw anything but ducks on the Loch that morning.
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Suppose that your prior belief in $A$ is $0.99$. Evaluate the probability that Nessie was out for a swim last Tuesday given your belief in MacWaffle's honesty. Now evaluate your (updated) level of confidence in $A$ (MacWaffle's honesty) given the claimed monster sighting.
The next week, MacWaffle reports that he saw the monster again on Wednesday! Recompute the probability that Nessie was out for a swim the second time using your updated estimate of MacWaffle's credibility. Finally, compute a second update of your level of confidence in MacWaffle after the second sighting.
Describe your conclusions in a brief paragraph. What would a good Bayesian conclude in the case of an (apparently) reliable witness who claims to see an (apparantly) very unlikely event? Draw conclusions for your day-to-day life. Note: Consider the possibility that you are wrong about your estimate of the probability of the ``unlikely'' event.
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