Exam 2 Sample Questions. Material for Exam 2 comes from Chapter G.1, G.2, G.3, and 14.1, 14.2

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1 Math 1620 Exam 2 Sample Questions Material for Exam 2 comes from Chapter G.1, G.2, G.3, and 14.1, 14.2 The exam will have 4 sections: Matching, Multiple Choice, Short Answer and one Logic Problem. A list of words that will be in the Matching section are provided below. Sample questions for the Short Answer section and a sample Logic Problem are included here. Matching Know the definitions of the following words for this section. You will be asked to match the word with the definition. Contrapositive Converse Expected Value Fallacy Implication Inverse Optimal Mixed Strategy Preference Table Reduction by Dominance Saddle Point Sound Argument Statement Strictly Determined Game Tautology Two-player Zero Sum Game Valid Argument Short Answer Questions: For calculations, show how you set-up each problem to receive full credit. 1. Reduce the payoff matrix by dominance. Row Player R1 R2 R Set up the payoff matrix for the following scenario: You are deciding whether to invade Finland, Sweden, or Norway, and your opponent is simultaneously deciding which of these three countries to defend. If you invade a country that your opponent is defending, you will be defeated (payoff: -1), but if you invade a country your opponent is not defending, you will be successful (payoff: +1). 3. Determine the optimal pure strategies under the minimax criterion for both players Row Player R1 R2 R

2 4. Find the expected value of the game with the following payoff matrix and using the mixed strategies S = [ 1 4, 1 4, 1 4, 1 4 ] and T = [ 1 4, 1 4, 1 4, 1 4 ]T. P = Decide which pure strategy (or strategies) the row player should use in order to maximize the expected value game with the following payoff matrix if the column player uses the strategy T = [ 1 4, 1 4, 1 2 ]T. P = Consider the following payoff matrix: Row R1 PlayerR Find the optimal strategies for the two players AND the value of the game. To receive full credit you must plot the strategy lines for each of the s strategies and use this to determine your optimal p value. 7. Construct a truth table for the statement p ( q r). 8. Complete the following truth table. p q p q p q T T T F F T F F 9. Given the statement: Married people are healthier than single people and more economically stable than single people, and children of married people do better on a variety of indicators. (a) Write the statement in symbolic form. Assign letters to simple statements that are not negated. (b) Construct a truth table for the symbolic statement in part (a). 10. Construct a truth table for [(p q) (q p)] (p q)

3 11. Given the statement: If you do homework right after class then you will not fall behind, and if you do not do homework right after class then you will. (a) Write the statement in symbolic form. Assign letters to simple statements that are not negated. (b) Construct a truth table for the symbolic statement in part (a). 12. Write the converse, inverse and contrapositive of the statement: If the stereo is playing, then I cannot hear you. 13. Use De Morgan s laws to write a statement that is equivalent to: It is not the case that my high school encouraged creativity and diversity. 14. Write the negation of the statement: A bill becomes law and it does not receive majority approval. 15. Use a truth table to determine whether the symbolic form of the argument is valid or invalid. p q q r p r 16. Translate the following argument into symbolic form. Then determine whether the argument is valid or invalid. Show your truth table or the standard valid or invalid form that you used in your determination. If we close the door, there is less noise. There is less noise. We closed the door. 17. Use Euler diagrams to determine whether the following arguments are valid or invalid. (a) (b) All dancers are athletes. Savion Glover is an athlete. Therefore, Savion Glover is a dancer. No blank disks contain data. Some blank disks are formatted. Therefore, some formatted disks do not contain data. 18. The theater society members are voting for the kind of play they will perform next semester: a comedy (C), a drama (D), or a musical (M). Use the following preference table to answer the questions below. Number of Votes First Choice M C D C D M Second Choice C M C D M D Third Choice D D M M C C (a) How many students voted? (b) Which type of play is selected using the plurality method?

4 (c) Which type of play is selected using the Borda count method? (d) Which type of play is selected using the plurality-with-elimination method? (e) Which type of play is selected using the pairwise comparison method? 19. In one sentence each, describe each of the four fairness criteria. 20. The preference table gives the results of a straw vote among three candidates: A, B, and Number of Votes First Choice C B A A C. Second Choice A C B C Third Choice B A C B (a) Using the plurality-with-elimination method, which candidate wins the straw vote? (b) In the actual election, the four voters in the last column, who voted A, C, B, in that order, change their votes to C, A, B. Using the plurality-with-elimination method, which candidate wins the actual election? (c) Is the monotonicity criterion satisfied? Explain your answer.

5 Logic Problem: Solve the following Logic Puzzle. When the Keebler family did their weekly shopping at Giant Grocery yesterday, Mrs. Keebler allowed each of the five children, including Lara, to pick a salty snack and a sweet snack for the coming week. No two Keebler children picked the same product. Given the shopping list that follows, you should be able to solve for the salty snack (one child chose Fritos Corn Chips) and sweet treat each child chose. 1. Ian helped his younger brother get his choice of Oreo Cookies from the top shelf. 2. The two Keebler girls decided they would share their snacks during the week and decided on Rold Gold Pretzel Rods and one of the two chip products chosen as their salty snacks. 3. The child who chose the Pizza Combos filled pretzels isn t the one who selected the Pepperidge Farm Tahoe Cookies. 4. Jeremy isn t the Keebler child who picked the Hostess Ho Ho cakes. 5. Neither the child who decided on the Oreos nor the one who chose the Little Debbie s Christmas Tree Cakes picked the Cheez-It Crackers to go with it. 6. The two girls selected Chips Ahoy Chocolate Chip Cookies and one of the two cake products chosen as their sweet treats. 7. The child who chose the Wavy Lay s Potato Chips, who isn t Kim, didn t pair it with the Little Debbie s cakes. 8. Neither Jeremy nor Micah chose the Tahoe Cookies. 9. Ian s choice of salty snack was neither the Cheez-Its nor the Wavy Lay s chips.