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\begin{center}\large 
MATH 245, Spring 2014

{\sc Homework 4}

due 10:45{\sc am} on Wednesday, April 2.
\end{center}

\noindent
{\em Background reading:} Sections 1.5, 5.1, and 5.3A.  
\begin{itemize}
\item About Markov chains: \\ \url{http://www.sosmath.com/matrix/markov/markov.html}
\item About Random Walks: \\ \url{http://www.ceid.upatras.gr/courses/probweb/lectures/lecture26.ps}
\end{itemize}

\smallskip\noindent
I ask that you do not contact previous Math Modeling students when completing this assignment.  Provide details of calculations and assertions that you include.  Don't forget to provide acknowledgments for those who helped you with the assignment and those resources that you consulted.

\begin{enumerate}
\item[\bf 4-1.] (4 pts)  Consider two events $E_1$ and $E_2$ that are {\bf dependent}  events. 

Will $\mathbb{P}(E_1 \textbf{ and } E_2) \geq \mathbb{P}(E_1)\mathbb{P}(E_2)$ or will $\mathbb{P}(E_1 \textbf{ and } E_2) \leq \mathbb{P}(E_1)\mathbb{P}(E_2)$? 

Determine all that you can. 

[{\em You may wish to consider multiple cases, perhaps investigating using Venn diagrams.}]

\item[\bf 4-2.] (4 pts) {\bf Determine the reliability of a computer}, where five types of components are required for printing out your group's project: {\em power}, {\em desktop}, {\em input}, {\em drive}, {\em printer}.  The {\em power supply} works $99.6\%$ of the time and the {\em desktop} works $99.9\%$ of the time.  There are two types of {\em input}, one of which must work: a keyboard with reliability $99.99\%$ and a mouse with reliability $99.98\%$.  There are three types of {\em drives}, one of which must work: the hard drive with reliability $99.7\%$, the CD-ROM with reliability $99.2\%$, and the flash drive with reliability $98\%$.  Last, there are two connected {\em printers}, one of which must work for success: a laser printer with reliability $99.5\%$ and a color printer with reliability $99\%$.  

Last, write a sentence or two discussing the assumptions that are required for your answer and calculations to be correct.

\item[\bf 4-3.] (6 pts) Do Exercise 1.5.2 on page 50 of our textbook.  Complete parts (a), (b), and (c) as given in Exercise 1.5.1.

\item[\bf 4-4.] (6 pts)  Suppose that you are setting up a pizza delivery business with three stores, Alpha Pizza in Flushing, Beta Pizza in Long Island City, and Gamma Pizza in Jamaica.

 When a customer calls a store, that store sends out a delivery person, who delivers the pizza and then returns to the closest store. Because of this, the delivery people end up transitioning from store to store throughout the night. 

\begin{itemize} 
\item[---]Suppose that when Alpha Pizza is called, then 1/2 of the time the delivery person returns to Alpha, 1/5 of the time the delivery person goes to Beta, and 3/10 of the time goes to Gamma. 

\item[---]When Beta Pizza is called, then 3/5 of the time the delivery person returns to Beta and with probability 1/5 the person goes to each of Alpha or Gamma. 

\item[---]When Gamma Pizza is called, then 3/5 of the time the delivery person returns to Gamma, 4/15 of the time goes to Alpha, and 2/15 of the time goes to Beta. 
\end{itemize}

\begin{enumerate}
\item Set up a Markov Chain model to simulate this situation.
\item Suppose that one evening there are 5 delivery people at each store at the beginning of the evening and they are all sent out at the same time.  What is the expected distribution of the delivery people when they return from this first delivery?
\item Determine the equilibrium distribution of the delivery people at the end of the day.
\end{enumerate}

\end{enumerate}


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