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1 1324 Test 1 Review Page 1 of 10 Review for Exam 1 Math 1324 TTh Chapters 7, 8 Problems 1-10: Determine whether the statement is true or false. 1. {5} {4,5, 7}. 2. {4,5,7}. 3. {4,5} {4,5,7}. 4. {4,5} {4,5,7} {3,5,7,9}. 5. {4,5} {4,5,7} {3,5,7,9}. 6. {4,5, 7} {4,5, 7}. 7. {1,3,5} {2, 4, 6}. 8. {4,5, 7} {7,5, 4}. 9. A B Afor all sets A and B. 10. A B Afor all sets A and B. 11. List all the subsets of { π,, }. 12. How many subsets does the set {Dennis, Ray, Bob, Pete, Jim, Darryl} have? 13. The management of Budget Fitness used a questionnaire to determine whether its members spent their gym time lifting weights, using cardio machines, or participating in group exercise classes. They found that 323 members lifted weights and 412 used the cardio machines, while 203 used both weights and cardio machines. 130 members participated in group exercise classes, and nearly all of these, 99 members, used cardio machines in addition to their exercise classes. Only 45 of the group exercise participants also did weightlifting. 30 of these 45 also worked out on cardio machines. There are 407 men and 245 women who have memberships at Budget Fitness, and children are not allowed to purchase memberships. HINT: Draw a Venn diagram there s room on the next page. a) How many members used only weights? b) How many of the gym s members do not use the weights, cardio machines, or group classes? (Budget Fitness has basketball, swimming, and other activities available.) c) How many members participated in group exercise classes but did not use any of the machines? d) If a member is selected at random, what is the probability that he or she uses cardio machines? e) If a member is selected at random, what is the probability that he or she uses cardio machines but not weights?

2 1324 Test 1 Review Page 2 of 10 In problems 14-17, use a Venn diagram to illustrate the indicated set. 14. A B 17. A' B ' 15. A B C 18. A' ( B C) 16. A' B 19. ( A' B) C'

3 1324 Test 1 Review Page 3 of Use a Venn diagram to illustrate a situation in which A B = and C B. 21. Use a Venn diagram to illustrate a situation in which C A B but C A B. 22. Use a Venn diagram to illustrate a situation in which A B B but C A B. 23. At a certain horse show, exhibitors are divided into professionals and amateurs. Of the professionals, 53% rode Western and the rest rode English. Of the amateurs, 77% rode Western and the rest rode English. 63% of the exhibitors are professionals. a) What is the probability that a randomly selected exhibitor rode English? b) Given that a randomly selected exhibitor rode Western, what is the probability that he or she is a professional?

4 1324 Test 1 Review Page 4 of A single card is drawn from a standard deck of playing cards. a) What is the probability that the card is a black seven? b) What is the probability that the card is higher than a five? (Assume aces high, outranking kings.) c) Given that the card is a face card, what is the probability it is a black queen? d) Given that the card is a heart, what is the probability that it is red? e) Test the events of drawing a face card and drawing a jack for independence. You must support your argument mathematically, showing how you performed the tests, and why you drew your conclusion. State your conclusion clearly. f) Find the odds against drawing a queen. 25. Suppose that two fair dice are rolled. a) What is the probability of rolling a sum less than 10? b) What are the odds for rolling a pair (both dice the same)?

5 1324 Test 1 Review Page 5 of Suppose that PA= ( ) 0.35, PB ( ) = 0.67, and PA ( B) = a) Determine PA ( B). b) Determine PA ( ' B). c) Determine PAB. ( ) 27. Suppose that a vase contains three red marbles and eight blue marbles. Two marbles are drawn in succession without replacement. a) What is the probability of drawing a blue marble on the second draw? b) What is the probability of drawing a blue marble on the second draw, given that a red marble was drawn on the first? c) What is the probability of drawing a blue marble on the first draw, given that a red marble was drawn on the second?

6 1324 Test 1 Review Page 6 of Suppose that two cards are drawn in succession from a standard card deck without replacement. a) What is the probability of drawing a heart on the first draw and a club on the second? b) Given that a club is drawn on the second draw, what is the probability that a heart is drawn on the first? 29. Suppose that events A and B are independent and that PA= ( ) 0.37 and that PB ( ) = What is PA ( B)? 30. Suppose that the ages of students in a certain college are described by the following table. Age Men Women < > a) What is the probability that a randomly selected student is over 40? (Give a decimal approximation accurate to the nearest thousandth.) b) What is the probability that a randomly selected male student is 30 or less? (Give a decimal approximation accurate to the nearest thousandth.)

7 1324 Test 1 Review Page 7 of Suppose that employees are given three attempts to pass a certain certification test. 82% of the employees pass it on their first attempt. Of those who fail the first attempt, 67% pass it on their second attempt. Of those who fail twice, only 27% pass it on their third try. a) What percentage of employees fail the test all three times? b) What percentage of employees fail the test twice but pass on the third try? 32. What would be an appropriate sample space for the experiment of tossing two coins? 33. Let x represent the number of heads when three fair coins are tossed. a) Write the probability distribution in the form of a table. b) Draw a histogram to represent the probability distribution. c) Determine the expected value of x.

8 1324 Test 1 Review Page 8 of Find the expected value of the number of girls in a family of 5 children, assuming boys and girls are equally likely. 35. Suppose an insurance company has determined that, in a given year, a home in a particular location has a probability of a loss causing the insurance company to pay the homeowner $30,000 and a probability of a loss causing a $10,000 payout. If the yearly premium is $2400, what is the expected value of the insurance policy to the customer? 36. Chili s Restaurant has a Create Your Own Combo special in which you choose your two favorite entrees from the following list: Grill Salmon w/ Garlic & Herbs, Half Rack of Baby Back Ribs, Margarita Chicken, Monterrey Chicken, Spicy Garlic & Lime Grilled Shrimp, and Chili's Classic Sirloin. a) How many possible Combos can be created? b) If you just can t decide what sounds good, and you select the combo randomly, what is the probability that your combo includes the salmon? c) Chili s also has a Triple Dipper special in which you choose 3 of 9 appetizers for $9.99. How many possible Triple Dippers can be created? 37. In how many ways can a quality control expert select a sample of 5 bolts for testing from a batch of 120? 38. Three cards are drawn from a standard card deck. What is the expected number of hearts?

9 1324 Test 1 Review Page 9 of Suppose a math teacher puts her students names in a hat and draws four of them to select students to put a homework problem on the board. If there are 27 students in the class, in how many ways can the problem-presenters be chosen? 40. Suppose that five people are running for the office of mayor in a nonpartisan election. It is possible that the person listed first on the ballot could have an advantage. To eliminate this possible bias in the ballots, the election committee has decided that an equal number of ballots should be printed with each possible arrangement of the five names. How many different ballots need to be created? 41. A 5-card hand is drawn from a standard card deck. a) What is the probability that it contains 3 hearts and 2 diamonds? b) What is the probability that all cards in the hand are hearts? c) What is the probability that all cards in the hand are of the same suit? (This is called a flush.) d) What are the odds against receiving a hand with all cards of the same suit? e) What is the probability that the hand contains exactly 2 aces and 2 kings? f) What is the probability that the hand contains 3 cards of one rank and 2 of another? (This is known as a full house. Examples would be the hand containing 2 sevens and 3 kings, or the hand containing 3 eights and 2 fours.)

10 1324 Test 1 Review Page 10 of Suppose that a batch of 120 bolts contains 8 that do not meet required tolerances for length, diameter, and thread spacing. An engineer selects a sample of 10 for testing. a) What is the probability that all 10 bolts in the sample meet required tolerances? b) What is the probability that the sample contains fewer than four bolts that do not meet required tolerances? c) What is the probability that the sample contains at least one bolt that does not meet required tolerances? 43. At a certain research conference, a random drawing is held in which graduate students receive a free night s hotel stay. There are 53 graduate students, and 39 of them are women. If 5 free hotel vouchers are given away what is the probability that there will be at least two men among the winners? 44. A basketball team has 12 players. At any one time, five of them are on the court, playing the distinct positions center, point guard, shooting guard, small forward, and large forward. a) How many possibilities are there for a group of five players to be on the court? b) How many different lineups are possible, taking into account the five different positions? c) How many different lineups are possible, taking into account the five different positions, if Yao Ming must play center?

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