# Math 160 - Projects

Listed below are the various projects that will be assigned throughout the semester. These projects take you above and beyond the material covered in the book or require outside data acquisition.

Each of these projects is worth 20 points and is due two class periods after the scheduled lecture for the corresponding chapter is finished. Due dates are noted on the class calendar.

You are encouraged to work in groups of up to three people per project (with the exception of the first part of the first project, which is an individual project). Turn in one project with all group member's names on it.

Plan on reading the section of the book dealing with the matter before we cover it in class; you will not always have time to finish the project if you wait until we do.

Some of these projects are very similar to problems that will appear on your exam. So even though they may not be due until after the exam, it would be wise for you to work and understand them before the exam.

## Project 1, Chapter 3

### Part I (10 points) - Individual

Your project is to plan a retirement fund for yourself. To simplify calculations, assume all transactions - starting of annuity fund, retirement, and death - occur on your birthday. Assume a nominal interest rate of 3% has been guaranteed for the remainder of your life. There is a financial planning worksheet to give you an idea of the format I'm looking for. Feel free to use that paper to write your answers on.

1. Identify the age you will be on your birthday this year.
2. Identify the age at which you wish to retire. Identify the number of years before retirement.
3. Identify the age at which you anticipate dying. Identify the number of years of retirement.
4. Identify the monthly payment you anticipate needing during your retirement.
5. Calculate the present value necessary on the date of retirement to finance your retirement.
6. The present value needed to retire is the future value necessary upon retirement. Calculate the monthly payment needed before retirement to have enough money to retire.
7. Calculate the amount of money in your retirement fund after ten years assuming you make the regular payments just calculated.
8. After the ten years, assume that you receive an inheritance of \$20,000 and add it to your retirement fund. If you stop making regular payments, and just let what money is in the account draw interest, what will the amount be at the time of retirement?
9. Subtract this amount from the future value needed upon retirement and recompute the monthly payment necessary to obtain the future value. Remember that ten years have gone by. If no more monthly payments are needed, then state the monthly benefit when you retire.

### Part II (10 points) - Group

Plan a house mortgage. Monthly payments will be made for 30 years on a fixed loan rate of 6%. Assume that you make a 20% downpayment.

1. Find a house you would like to purchase that costs between \$80,000 and \$200,000. Include the address and cost of the home in your project. You can find homes in the newspaper or online at http://www.realtor.com/decaturil/
2. How much is the down payment?
3. Compute the monthly payment needed to finance the balance of the house.
4. How much will money will you pay to repay the loan?
5. How much interest will you pay?
6. It is now ten years later and you have made regular payments on the house.
1. Determine how much money is still owed on the house after 10 years.
2. How much is the house worth if it is now worth 8% more than what you paid for it?
3. How much equity do you have in the house?
7. Going back to when you took out the loan, assume that you pay an extra \$250 a month.
1. How long will it take to pay off the loan?
2. How much will you repay
3. How much interest will you pay?

## Project 2, Chapter 4

### Part I (10 points)

Solve the following matrix equations for X if possible. If it canot be solved, write "not possible". Assume capital letters represent matrices and any necessary inverses exist.

1. AX = B
2. AX + BX = C
3. XA + XB = C
4. AX - X = B
5. AX - 3X = B
6. XA - 3X = B
7. AX + B = CX + D
8. X = MX + D
9. AXA = B
10. AX + XA = B

### Part II - The Internet2 (10 points)

The Internet2 is a high-speed network for the U.S. research and education community. The network is connected through Internet2 Router sites, located in 9 cities throughout the United States. There is a map of the Internet2 network at http://www.internet2.edu/pubs/networkmap.pdf

There is a blank incidence matrix that you may wish to print and use to make this easier for you. In the following, "hop" or "network segment", means a trip between two cities.

1. Create an incidence matrix for the Internet2 router sites (all traffic is bi-directional). List the cities in alphabetical order.
2. What is the maximum number of hops a packet might require before reaching its destination? (Hint: find A+A2+A3+…+An until every non-diagonal element is greater than zero). Write down this matrix.
3. Part of the design of a good network is redundancy. If any one site loses connectivity, the rest of the network must continue to function. If the site at Kansas City goes down, what is the maximum number of hops a packet might require before reaching its destination (assume its destination isn't Kansas City)? Write down the matrix used to determine this answer.

I also strongly recommend using Maxima or similar computer program to find your answers. If you don't want to write the answers down by hand into the incidence matrices, you can copy the output from Maxima into Word and then print that. However, you will need to manually label the rows and columns if you do that.

## Project 3, Chapter 8

### Part I - Poker Probabilities (5 points)

Find the probabilities of the following winning poker hands. Assume aces are high and that five cards are drawn from a standard 52 card deck. Show work and the probabilities.

1. Four of a Kind
2. Full House - (Three of one kind of card and two of another)
3. Three of a Kind
4. Two Pairs
5. One Pair

### Part II - Illinois Lottery (5 points)

Use the information on the Illinois Lottery website http://www.illinoislottery.com to answer this part. The rules and the chances of winning are given on the website. The chances of winning are given on the website, so it is important that you show the work necessary to arrive at the answer. Find the probabilities that a ticket will win the following prizes. (Hint: Think of the Mega Millions game as a combination of two separate games, each using a hypergeometric distribution).

1. Mega Millions game, second prize
2. Mega Millions game, third prize
3. Mega Millions game, fourth prize
4. Lotto, second prize
5. Lotto, third prize

### Part III - Decision Theory (10 points)

John and Mitchy run a computer store. They can purchase 10 computers from Zol and Denny for \$1400 each, 30 computers from McGuinn and McGuire for \$1300 each, or 50 computers from Sebastian for \$1250 each (they can buy from more than one dealer, but only one order per dealer). John and Mitchy sell the computers for \$1500 each. Each computer that is left at the end of the month will be sold in a clearance sale for \$900. John and Mitchy estimate a loss of goodwill of \$50 for each customer which comes into the store, but is unable to purchase a computer. During the month, the customers will either demand 15, 30, 45, or 60 computers. Assume the probability of 15, 30, 45, or 60 computers is 0.10, 0.15, 0.50, and 0.25 respectively.

1. Create a payoff table with the five actions (remember that you can combine purchases from more than one dealer and some plans don't make any sense when the demand is considered) and four states of nature (demand)
2. Create the opportunistic loss (regret) table.
3. For each decision criteria (expected value, maximax, maximin, minimax), find the payoff or loss for each action and the best action.

## Project 4, Chapter 9

### Part I - Peg Moving (10 points)

A game is played by placing a peg into one of ten holes arranged as shown in the figure. The peg is then randomly moved to one of the adjacent holes (as an example, F is adjacent to C, E, I, and J) until one of the vertices (A, G, or J) is reached. A peg moving worksheet has been created to help you setup the problem.

1. Create a transition matrix.
2. Find the fundamental matrix F.
3. If the game is begun by placing the peg into hole E, how many moves can be expected to be made before the game is over?
4. What is the probability of ending up in hole A if the peg is placed into hole F to begin with?

### Part II - Craps (10 points)

Consider the dice game of craps as an absorbing Markov chain. The rules of craps are as follow: A pair of dice are rolled. If the sum on the first roll is a 7 or an 11, you win immediately and the game is over. If the sum on the first roll is a 2, 3, or 12, you lose immediately and the game is over. If the sum on the first roll is a 4, 5, 6, 8, 9, or 10, that sum becomes the "point" and you continue rolling the dice until you roll your point again and win or you roll a 7 and lose.

A craps worksheet that can be completed is available in PDF format. If you complete the last two pages of the worksheet, then you have answered all the questions below.

1. Write a 1×3 matrix indicating the probabilities of winning, losing, or making a point on the first roll of the dice.
2. Find the expected number of rolls before winning or losing and the probability of winning and losing for each of the point values.
3. Find the overall probability of winning and losing a game of craps. Note that the probabilities of winning and losing for the individual point values are conditional probabilities ... they are dependent upon rolling that particular point value on the first roll, so use the general multiplication rule.
4. Find the expected number of rolls for a game of craps. To find the expected number of rolls for a point (since they're different depending on the point value), find the expected value of the number of rolls when weighted with the probability of rolling that point. Be sure and add one to the number of rolls to include the first roll of the dice.

## Project 5, Chapter 10

### Part I (10 points)

Consider the following two-person zero-sum game.

Cereal Dairy Meat Snacks -3 1 -5 4 2 -1 -2 3 -1 1 3 -2 3 4 -1 2

The row player is "Rick's Ready Mart" and the column player is "Corissa's Country Market".

Rick and Corissa own the only two grocery stores in town so that a sale for Rick is a loss for Corissa and vice versa. Each week, they each run a special on exactly one type of food in an effort to draw business into their store. The matrix showing the choices and the gain in sales for Rick's store are shown.

1. If Rick and Corissa each randomly select a food type to put on sale, what are the strategies and what is the value of the game for Rick?
2. What are the optimal strategies for Rick and Corissa? What is the value of the game for Rick under that those strategies?
3. Find the expected payoff values for each of Rick's actions if Corissa plays her optimal strategies. Find the expected loss values for each of Corissa's actions if Rick plays his optimal strategies.
4. Rick finds out that Corissa is going to spin the spinner from the game Life (10 slots) and place Cereal on sale if a 1 shows up, Dairy on sale if a 2 or 3 shows up, Meat on sale if a 4, 5, or 6 shows up, and Snacks on sale if a 7, 8, 9, or 10 shows up.
1. What is the expected value of each action for Rick?
2. What should Rick's a priori strategy be using the expected value criterion?
3. What should Corissa's strategy really be (not what she said) if the intent was to trick Rick into playing a particular strategy?

### Part II (10 points)

1. Create a 4 by 4 non-strictly determined game matrix with no recessive rows or columns.
2. Turn the matrix into a story problem. The matrix can be given as a matrix, but come up with choices for the row and column players to make it an interesting problem.
3. Solve the game using the calculator.