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Section 13.1

Chapter 13. Section 13.1. Exercise #1. The preference ballots for the election of a CEO by the board of directors are shown. Make a preference table for the results of the election and answer each question. How many people voted?

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Section 13.1

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  1. Chapter 13 Section 13.1 Exercise #1

  2. The preference ballots for the election of a CEO by the board of directors are shown. Make a preference table for the results of the election and answer each question. How many people voted? How many people voted for candidates in the order of preference XZY? How many people voted for candidate Y as their first choice? Using the plurality method, determine the winner of the election.

  3. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes First choice X Second choice Y Third choice Z

  4. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes First choice X Second choice Y Third choice Z 5

  5. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes 5 First choice X X Second choice Y Z Third choice Z Y

  6. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes 5 First choice X X Second choice Y Z Third choice Z Y 4

  7. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes 5 4 First choice X X Y Second choice Y Z Z Third choice Z Y X

  8. Number of votes 5 4 First choice X X Y Second choice Y Z Z Third choice Z Y X X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X 8

  9. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes 5 4 8 First choice X X Y Z Second choice Y Z Z Y Third choice Z Y X X

  10. X Y Z X Z Y Y Z X Z Y X X Y Z Y Z X Z Y X Z Y X X Y Z Y Z X X Z Y Y Z X X Y Z Y Z X X Z Y Y Z X Z Y X Z Y X X Y Z X Z Y Y Z X Y Z X Number of votes 5 4 8 First choice X X Y Z Second choice Y Z Z Y Third choice Z Y X X 5

  11. 5 4 8 5 Number of votes First choice X X Y Z Second choice Y Z Z Y Third choice Z Y X X a) How many people voted? 5 + 4 + 8 + 5 = 22 22 people voted in the election.

  12. 5 4 8 5 Number of votes First choice X X Y Z Second choice Y Z Z Y Third choice Z Y X X b) How many people voted for candidates in the order of preference XZY? 4 people voted in the order XZY.

  13. 5 4 8 5 Number of votes First choice X X Y Z Second choice Y Z Z Y Third choice Z Y X X c) How many people voted for candidate Y as their first choice? 8 people voted for candidate Y as their first choice.

  14. 5 4 8 5 Number of votes First choice X X Y Z Second choice Y Z Z Y Third choice Z Y X X d) Using the plurality method, determine the winner of the election. X received 5 + 4 or 9 Winner! Y received 8 Z received 5

  15. Chapter 13 Section 13.1 Exercise #5

  16. Students at a college were asked to rank three improvements that they would like to see at their college. The choices were build a new gymnasium (G), build a swimming pool (S), or build a baseball / football field (B). The votes are summarized in the preference table.

  17. How many students voted? What option won if the plurality method was used to determine the winner?

  18. 83 56 42 27 Number of votes First choice G S S B Second choice S G B S Third choice B B G G a) How many people voted? 83 + 56 + 42 + 27 = 208 208 students voted.

  19. 83 56 42 27 Number of votes First choice G S S B Second choice S G B S Third choice B B G G b) What option won if the plurality method was used to determine the winner? G received 83 S received 56 + 42 or 98 B received 27

  20. 83 56 42 27 b) What option won if the plurality method was used to determine the winner? Number of votes First choice G S S B Second choice S G B S Third choice B B G G The option to build a new swimming pool (S) won with 98 votes.

  21. Chapter 13 Section 13.1 Exercise #7

  22. Students in a sorority voted to select a floral bouquet for their representative in a homecoming parade. The choices were roses (R), gardenias (G), carnations (C), and daisies (D). The preference table is shown.

  23. How many students voted? What flower bouquet won if the plurality method was used to determine the winner?

  24. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G a) How many people voted? 3 + 5 + 2 + 6 + 4 = 20 20 students voted.

  25. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G b) What flower bouquet won if the plurality method was used to determine the winner? R received 3 + 4 or 7 D received 5 C received 2 + 6 or 8

  26. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G b) What flower bouquet won if the plurality method was used to determine the winner? Carnations (C) were selected for the parade with 8 votes.

  27. Chapter 13 Section 13.1 Exercise #9

  28. Using the election results given in Exercise 5, has the head-to-head comparison criterion been violated? Explain your answer.

  29. 83 56 42 27 Number of votes First choice G S S B Second choice S G B S Third choice B B G G Column 1: G was preferred over S by 83 votes.

  30. 83 56 42 27 Column 1: G was preferred over S by 83 votes. Number of votes Column 2: First choice G S S B Column 3: Second choice S G B S Column 4: Third choice B B G G S was preferred over G by 56 + 42 + 27 or 125 votes. 125 > 83 S wins over G.

  31. 83 56 42 27 Column 1: S was preferred over B by 83 + 56 + 42 or 181 votes. Column 2: Number of votes Column 3: First choice G S S B Second choice S G B S Third choice B B G G

  32. 83 56 42 27 Column 1: S was preferred over B by 83 + 56 + 42 or 181 votes. Column 2: Number of votes Column 3: First choice G S S B B was preferred over S by 27 votes. Second choice S G B S Third choice B B G G Column 4: 181 > 27 S wins over B.

  33. 83 56 42 27 Column 1: G was preferred over B by 83 + 56 or 139 votes. Column 2: Number of votes First choice G S S B Second choice S G B S Third choice B B G G

  34. 83 56 42 27 Column 1: G was preferred over B by 83 + 56 or 139 votes. Column 2: Number of votes First choice G S S B Column 3: B was preferred over G by 42 + 27 or 69 votes. Second choice S G B S Column 4: Third choice B B G G 139 > 69 G wins over B.

  35. 83 56 42 27 Number of votes First choice G S S B Second choice S G B S Third choice B B G G S won by the plurality method. S wins over both G and B in head-to-head comparison. Thus, the head-to-head criterion has not been violated.

  36. Chapter 13 Section 13.1 Exercise #11

  37. Using the election results given in Exercise 7, has the head-to-head comparison criterion been violated? Explain your answer.

  38. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G Column 2: Column 3: C was preferred over R by 5 + 2 + 6 or 13 votes. Column 4:

  39. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G Column 2: Column 3: C was preferred over R by 5 + 2 + 6 or 13 votes. Column 4: Column 1: R was preferred over C by 3 + 4 or 7 votes. Column 5: 13 > 7 C wins over R.

  40. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G Column 2: Column 3: C was preferred over G by 5 + 2 + 6 + 4 or 17 votes. Column 4: Column 5:

  41. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G Column 2: Column 3: C was preferred over G by 5 + 2 + 6 + 4 or 17 votes. Column 4: Column 5: G was preferred over C by 3 votes. Column 1: 17 > 3 C wins over G.

  42. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G Column 1: Column 3: C was preferred over D by 3 + 2 + 6 + 4 or 15 votes. Column 4: Column 5:

  43. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G Column 1: Column 3: C was preferred over D by 3 + 2 + 6 + 4 or 15 votes. Column 4: Column 5: D was preferred over C by 5 votes. Column 2: 15 > 5 C wins over D.

  44. Number of votes 3 5 2 6 4 First choice R D C C R Second choice G C R G C Third choice C G G D D Fourth choice D R D R G C won by the plurality method. C also won over all other choices in head-to-head comparison. Thus the head-to-head criterion has not been violated.

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