MAT 17: Introduction to Mathematics Final Exam Review Packet. B. Use the following definitions to write the indicated set for each exercise below:

Transcription

1 MAT 17: Introduction to Mathematics Final Exam Review Packet A. Using set notation, rewrite each set definition below as the specific collection of elements described enclosed in braces. Use the following definition to assist with the appropriate exercise(s): M = { January, February, March,, December } 1. A = { x x is a negative integer; x 3 } 2. B = { x x M; x has less than 31 days } B. Use the following definitions to write the indicated set for each exercise below: 3. X Z X = { 1, 2, 3, 4, 5 } Y = { 2, 4, 6, 8, 10 } Z = { 4, 8, 12, 16, 20 } 4. (Y Z) X

2 C. Use the following definitions to determine whether each statement is true or false. Be sure to write the entire word TRUE or FALSE as your final answer. A = { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z } B = { a, e, i, o, u, y } C = { a, e, i, o, u } 5. A and B are disjoint sets. 6. A C 7. B C = D. Using the information below, write the appropriate numerical value in each region of the Venn diagram provided. Be sure to place the proper value in each region based on the labels provided in the diagram students are surveyed to determine what days of the week they like. The results of the survey are as follows: 140 said they like Friday 150 said they like Saturday 160 said they like Sunday 110 said they like Friday and Saturday 115 said they like Friday and Sunday 120 said they like Saturday and Sunday 90 said they like all three days Friday Saturday Sunday

3 E. Using the definition of a logical statement, write YES or NO to indicate whether or not each of the sentences below is a logical statement. 9. I like apples. 10. Apples are the best fruit. 11. This sentence is false = 3 F. Write the negation of each statement below All animals have four legs. 15. No dogs have four legs. 16. Some dogs have four legs. G. Answer each question below. 17. What is the converse of the contrapositive of ~q p? 18. What is the inverse of the inverse of the inverse of the inverse of the converse of the contrapositive of the converse of the converse of p q? 19. How many variables are in a truth table that consists of 16 rows?

4 H. Determine whether each statement below is TRUE or FALSE based on the following conditions: 20. [ (p q) (~ r s) ] ~ p p = FALSE q = FALSE r = FALSE s = TRUE 21. (~ s ~ q) (~ p ~ r) 22. (~ p q) (~ r ~ s) I. Construct and complete your own truth table for the following logical argument (be sure to translate the argument into symbols first) and determine whether or not the argument is valid: 23. If it rains, then the ground will get wet. The ground did not get wet., It did not rain.

5 K. Complete each exercise below. Be sure to show all work to receive full credit. 24. How many distinct ways can the letters of the word LOLLIPOP be arranged? 25. How many ways can a committee of 5 people be selected from a group of 20 people? 26. A box of 20 bananas has five rotten bananas in it. If a sample of four bananas is selected at random, how many such samples contain less than 3 rotten bananas? 27. How many ways can a teacher line up 7 of her 22 students at the chalkboard for an exhilarating game of 7-up? 28. How many five-card poker hands contain three of a kind (3 of the same denomination and 2 different non-matching denominations)?

6 29. A closet contains 15 footballs, 12 baseballs, and 10 basketballs. How many ways can a person randomly pull out 6 balls in the following order: football-football-baseballbasketball-football-football. 30. How many ways can a person roll a total of 8 on one roll of two fair dice? 31. A spinner has the numbers 1 through 10 on it. How many ways can a person spin no 7s on 25 spins? 32. Find the number of ways to pick a random group of 10 marbles consisting of 3 red, 3 white, and 4 blue marbles out of a bag containing 5 of each color marble. 33. How many five-card poker hands consist of no diamonds?

7 34. How many ways can 12 tosses of a fair coin result in exactly 3 heads being flipped? 35. How many ways can a person flip a tail on one flip of a coin followed by rolling a prime number on one roll of a fair die followed by randomly choosing a weekday from the seven days of the week? 36. How many ways can a total greater than 3 on one roll of two fair dice occur? 37. The winning Powerball numbers are determined by randomly selecting 5 white numbered ping pong balls and 1 red numbered ping pong ball from a total of 59 white and 35 red numbered ping pong balls. What are the odds against winning the Powerball jackpot? 38. Determine the odds in favor of winning a race if the probability of winning is 35%.

8 39. The odds against passing a particular final exam are 3 2. What is the probability of failing that final exam? 40. Determine the probability of exactly 2 yellow peas being in a sample of 6 peas randomly selected from a bin containing 5 yellow and 10 green peas. 41. What is the probability of spinning an odd multiple of 3 on a standard 1-to-10 spinner, then rolling a prime number less than 4 on a fair die, then flipping a tail on a fair coin, and finally randomly selecting a face card from a standard deck of playing cards? 42. Each of the letters in the word "IOWA" is written on its own slip of paper, and the slips of paper are placed in a hat. Similarly, the letters in the word "MAINE" are placed in a second hat, and the letters in the word "UTAH" are placed in a third hat. One letter is randomly selected from the first hat, recorded, and placed in the second hat. Then, one letter is randomly selected from the second hat, recorded, and placed in the third hat. Finally, one letter is randomly selected from the third hat and recorded. Determine the probability that the first letter was a vowel, the second was a consonant, and the third was a vowel. 43. Determine the probability that the last letter selected in the experiment described in Exercise #42 above was a vowel.

9 44. An archer has a 10% chance of hitting a bull's eye on a target, a 30% chance of hitting the red ring surrounding the bull's eye, a 40% chance of hitting the blue ring surrounding the red ring, a 15% chance of hitting the black ring surrounding the blue ring, and a 5% chance of missing the target altogether. A bull's eye is worth 10 points, the red ring is worth 7 points, the blue ring is worth 5 points, the black ring is worth 3 points, and missing the target is worth 0 points. Determine the expected value of each arrow the archer shoots at the target. 45. A computer model indicates a pending snowstorm has a 20% chance of dropping 1 inch of snow, a 40% chance of dropping 2 inches of snow, a 30% chance of dropping 3 inches of snow and a 10% chance of dropping 4 inches of snow. What amount of snow accumulation should an observer of the snowstorm expect to see? 46. A carnival game consists of paying $5.00 to randomly select one card from a table on which 25 cards are arranged face down in a 5 5 grid. The contestant flips over the selected card and wins the dollar amount shown. If 8 cards are worth $1.00 each, 6 cards are worth $2.00 each, 4 cards are worth $3.00 each, 3 cards are worth $5.00 each, 2 cards are worth $10.00 each, 1 card is worth $20.00, and 1 card is worth $50.00, is this a fair game (i.e., is the expected value of the game equal to the price paid to participate)? If not, does the contestant or the game operator come out ahead, and by how much per draw? 47. Determine the probability of rolling a 5 exactly 3 times on 10 rolls of a fair die.

10 48. The three-point shooting percentage for a particular basketball all-star is 47.6%. Determine the probability that this all-star makes exactly 20 three-point shots in 50 attempts at a shooting exhibition % of Gizmos produced by ABC Corporation are defective. Determine the probability that a shipment of 1000 Gizmos contains exactly 25 defective Gizmos % of New Jersey drivers are better than all Pennsylvania drivers. What is the probability that a random sample of 100 New Jersey drivers contains less than 98 drivers who are better than all Pennsylvania drivers?