# Math 160: Study Guide - Chapters 9-10

1. Solve a strictly determined game. Show all work.
2. Eliminate any recessive rows and/or columns. Do NOT solve the game.
3. Create a transition matrix from the transition diagram.
4. Solve a 2×2 non-strictly determined game. Show all work. Know the formulas from section 10.2.
5. 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.
6. Setup a game problem and then use the calculator to find the solution. Look at problem 10.4.9-12.
7. The initial system to a linear programming problem is given. Complete the table of values, determine if the points are feasible or not, and give the optimal solution to the problem.
8. A final tableau from a game is given. Give the optimal row and column strategies and the value of the game.
9. Write the linear programming problems necessary to solve a game. Do not attempt to solve the game.
10. Markov chain problem. Fill in the missing probabilities in the transition matrix, give an initial state matrix and find a state matrix for future time period. Find the steady state matrix.
11. Bayesian probability problem. Complete the joint probability distribution and then use it to answer the questions.
12. Given P(A), P(B) and one more probability, complete a probability distribution and then find several probabilities from the probability distribution. Know how to tell whether two events are independent.
13. Finance application problem involving annuities.
14. Solve a 3x3 system of linear equations. Either use Gauss-Jordan elimination (pivoting or the rref calculator command) or use the inverse of a matrix.
15. Find the solution to a game using the calculator. Find the payoffs for the row player under the expected value, maximax, maximin, and minimax criteria. Find the value if both players play randomly (all choices are not equally likely - you need to have a concept of what a relative frequency is).
16. Find the number of ways a poker hand can occur. Use the "hypergeo" program. The deck has been modified, but the changes to the deck are given.
17. Absorbing markov chain. Find probabilities of a certain transition, how many transitions should be expected before reaching an absorbing state and the long term probability of reaching an absorbing state. Use the information on the take home portion of the exam to answer this question.
18. Finance application involving annuities.
19. Probability problem. Use the multiplication rule to find the probability of compound events occurring.

## Notes

• Problems 1-5 must be answered without a calculator. The game program on the calculator can be used for the rest of the test whenever you need to find an optimal strategy.
• Problems 6-16 should make full use of the calculator programs and other capabilities.
• The solution to a game consists of the optimal row strategy P*, the optimal column strategy Q*, and the value of the game v. Make sure you give all three parts, not just the value of the game.
• Move swiftly through the first portion of the test.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Total 3 3 3 5 10 9 8 10 10 10 10 20 9 3 20 12 15 6 9 175