# Math 160: Study Guide - Final Exam

1. Compound interest problem. Use the "finance" program.
2. Present value problem. Use the "finance" program.
3. Solve a 3x3 system of linear equations. Either use Gauss-Jordan elimination (pivoting) or use the inverse of a matrix.
4. Retirement problem. Figure out what the need to retire and what it will take each month to save that. Figure out how long the retirement will last if extra is paid. Find the remaining balance after a certain period of time. Use the "finance" program.
5. Give the solution to a system of linear equations from an augmented matrix in row echelon form.
6. Leontief input output problem. Use the "leontief" program.
7. The initial tableau from a non-standard maximization problem is given. Write the maximization problem.
8. Maximize a standard linear programming problem. Write the initial tableau and final tableau, and give the values of the objective function and decision variables. Use the "simplex" program.
9. Given P(A), P(B) and one more probability, complete a probability distribution and then find several probabilities from the probability distribution.
10. Find the mean, median, and sample standard deviation for a set of data.
11. Binomial probabilities. Use the "binomial" program.
12. Solve a game matrix, giving the optimal row and column strategies, and the value of the game. Use the "game" program.
13. Markov chain problem. Write the initial state matrix, the transition matrix. Find the first state matrix, and the steady state matrix.
14. Linear programming problem. A tableau is given. Label the columns as appropriate using x's for the decision variables, s's for the slack variables, z for the objective function, and rhs for the right hand side. Identify which variables are basic and which are non-basic and give their values. Identify the basic variable for each row of the tableau and find the appropriate ratios on the right side. Circle the pivot element. Identify which variable is entering and which is exiting.
15. Decision Theory. Most of the payoff table has been created for you, but you need to find two entries in the table. Create the opportunistic loss table. Then give the value and optimal action under each criteria.
16. Probability problem. Two bags with different types of coins in them. A situation is described and you need to draw a tree diagram and then find some probabilities.
17. The payoffs for a game are given. Find the probabilities of each payoff and then find the expected value. Identify whether the game is fair or not. You may use the "pdist" program, although it's probably easier to do by hand.
18. The final tableau from a zero-sum, two-player game is given. Give the optimal row and column strategies and the value of the game.
19. Absorbing Markov chain problem. Write the initial state matrix and the transition matrix. Then find the expected number of transient states before leaving the matrix.

## Notes:

• The test is open notebook. Make sure that your notes are complete in the sections covered on the exam.
• Problems 1-13 are to be worked individually. Problems 14-19 may be worked in a group of up to three people. You may not have the individual portion of the test with you while you are working in groups.
• There is no requirement to work in groups. If you decide to get into groups, you will not be able to use your notebook. If you decide to work alone, you may use your notebook.
• You may begin the group portion of the test alone with your notebooks and then give up your notebooks once you've exhausted your ability to answer the questions.
• The exam is laid out mostly in chapter order. Having your notes in order will help.
• The individual part of the exam is 126 points, the group part is worth74 points.

## Points per problem

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Tot. 6 6 6 12 6 6 6 16 16 9 9 12 16 12 12 14 14 10 12 200