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\title{\sc Math 60 -- Homework 7}
\author{due Wednesday, May 16}
\date{}
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\maketitle
\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Staple your work together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive full credit. Every step of your answers must be fully justified to receive credit.
%\emph{You may use Wolfram Alpha or another similar tool to compute any necessary sums or integrals, and for your matrix calculations. If you have trouble with this, let me know.}\\
%\textbf{If you're using facts about distributions to answer the questions, be very clear about which distribution you're using to model that problem and why that distribution is appropriate.}
\underline{\textbf{It is considered cheating and a violation of the Honor Code to look for answers to these problems on the internet.}}
\ \\
\framebox{\emph{You may use a $z$-table on any question.}}
\begin{enumerate}
\item What do Chebyshev's Inequality and the Law of Large Numbers say about the probability of getting at least 75 heads when flipping a fair coin 100 times? \emph{Hint:} Improve your bound by using the fact that the binomial distribution is symmetric.
\item A share of common stock in the Pilsdorff beer company has price $Y_n$ on the $n$th business day of the year. ($Y_n$ is a random variable.) Finn observes that the price change $X_n = Y_{n+1} - Y_n$ appears to be a random variable with mean $\mu = 0$ and variance $\sigma^2 = 1/4$. If $Y_1 = 30$, find a lower bound for the following probabilities, under the assumption that the $X_n$'s are mutually independent.
\begin{enumerate}
\item $P(25 \leq Y_2 \leq 35)$
\item $P(25 \leq Y_{11} \leq 35)$
\item $P(25 \leq Y_{101} \leq 35)$
\end{enumerate}
\item Each student's score on a particular calculus final is a random variable with values in the range $[0,100]$, mean $70$, and variance $25$.
\begin{enumerate}
\item Find the best lower bound you can (using only the tools we've learned), for the probability that a particular student's score will fall between $65$ and $75$.
\item If $100$ students take the final and the students' grades are all mutually independent, find a lower bound for the probability that the class mean will fall between $65$ and $75$.
\end{enumerate}
\item A fair coin is tossed 10,000 times.
\begin{enumerate}
\item Using a binomial distribution and Wolfram Alpha (or another similar tool) for experimentation, find the number $m$ such that the probability of flipping between $5000-m$ and $5000+m$ heads is closest to $2/3$.
\item Answer the same question, but using only the Central Limit Theorem and a $z$-table.
\end{enumerate}
\item Once upon a time, there were two railway trains competing for the passenger traffic of 1000 people leaving from Chicago at the same hour and going to Los Angeles. Assume that passengers are equally likely to choose each train and that their choices are mutually independent. How many seats must a train have to assure a probability of $0.99$ or better of having a seat for each passenger? (Use the Central Limit Theorem.)
\item A club serves dinner to members only. They are seated at $12$-seat tables. The manager observes over a long period of time that $95$\% of the time there are between six and nine full tables of members, and the remainder of the time the numbers are equally likely to fall above or below this range. Assume that each member decides to come with a given probability $p$ and that the decisions are mutually independent. How many members are in the club?
\end{enumerate}
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