2. Let E and F be two events of the same sample space. If P (E) =.55, P (F ) =.70, and


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1 c Dr. Patrice Poage, August 23, Exam 1 Review NOTE: This review in and of itself does NOT prepare you for the test. You should be doing this review in addition to all your suggested homework, quizzes, and worksheets. 1. Researchers asked students in grades 4 through 6 in three school districts in Michigan about what they thought was the most important thing in school. The results are below. Based on these results, what is the probability a person randomly selected from this group: (a) thinks grades are most important? (b) Is from an Urban area, given they think Sports is most important? (c) Thinks being Popular is most important, given they are from a Rural area? 2. Let E and F be two events of the same sample space. If P (E) =.55, P (F ) =.70, and P (E F c ) =.25, what is P (E F ) =? 3. Let E and F be mutually exclusive events of the same sample space. The P (E c ) =.3 and P (F ) =.15. What is P (E c F c )? 4. An experiment consists of rolling a red 4sided die and a blue 4sided die at the same time. (a) What is the sample space for this experiment? (b) What is the event, E, that sum of 6 is rolled? (c) What is the event, F, that the red dice rolled a 2? (d) Find P (F E) (e) What is the probability that a sum of 3 is rolled or the red dice rolled an odd number? (f) What is the probability that the sum of the dice is less than 5 and the blue die rolled a number less than 4? (g) What is the probability the sum is greater than 3, given that the red die rolled an even number. 5. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {6, 7, 8, 9}, and C = {1, 2, 3, 8, 9}. Find the following sets: (a) A c B c (b) B (A C) 6. If n(u) = 33, n(a B) = 29, n(a B) = 5, and n(b c ) = 23, Find n(a c ). 7. Shade the following Venn Diagram to represent B (A c C) 8. Shade the following Venn Diagram to represent A c (B c C) 9. A math prof suggests to her class of 34 students that they should decorate their graphing calculator covers. 4 students end up not decorating them at all. 17 students put stickers on the cover. 21 students used paint pens on the cover. How many students put stickers on and used paint pens?
2 c Dr. Patrice Poage, August 23, A local high school math teacher conducted a survey to find out the correlation between students who were suspended from school and whether or not they passed their math class. Fill in the Venn diagram below based upon the following results. NOTE: Two of the numbers have been filled in for you. 56 boys were surveyed 23 boys who got suspended still passed 53 students failed 44 girls did not get suspended 29 boys passed the class 8 boys were suspended and failed 11. In the previous problem, what does the number outside the circles, but inside the rectangle represent? 12. In New York there is no limit to the number of times one can take the Bar Exam. A group of New York lawyers were surveyed. Let the random variable, X, denote the number of times a lawyer took the Bar Exam until he/she passed. How many times can one expect to take this test before he/she passes it? 13. An experiment consists of tossing a coin and observing the side that lands up and then randomly selecting a marble from a jar filled with 2 red, 5 blue, and 1 yellow marbles and noting the color. What is the sample space for this experiment? 14. Let E and F be two events in the same sample space. Suppose P (E) = 0.38, P (E F ) c = 0.28, and P (E F ) = Find (a) P (F ) = (b) P (E c F ) = (c) P (E F c ) = (d) P (E c F c ) = 15. Two fair 6sided dice (# d 16) are rolled at the same time. One of the dice is blue and the other is red. What is the probability (a) the sum of the dice is odd or at least one dice rolled a 3? (b) the sum is greater than 5, if it is known that the blue die rolled a number great than 2? 16. A single card is drawn from a standard deck of 52. (a) What are the odds against drawing a spade? (b) What are the odds in favor of drawing a red face card?
3 c Dr. Patrice Poage, August 23, Based on the table below, what is the probability that a randomly selected person (a) will prefer history, given that the person is an 8th grader? (b) is a 7th grader given that the person prefers English? 18. The pie chart below shows the results of a survey taken among a group of high school students as to what their favorite Free Time activity is. If a student from this group is randomly selected, (a) Find the odds against the student chose Reading Books. (b) Find the odds in favor the student likes Playing Game, Listening to Music, or Watching TV. 19. A survey was conducted on 100 people who visited the local health clinic today. Fill in the Venn diagram completely, based upon the following results. (NOTE: 2 numbers have been filled in for you) 44 did not have a cough 35 had a sore throat and runny nose 57 had a runny nose 31 had a cough and runny nose 22 did not have a sore throat or runny nose cough 11 runny nose sore throat 10 U 20. Let U denote the set of all people on the campus of Texas A&M. Let F = {x x is a female} R = {x x has red hair} S = {x x is a student} (a) Describe the following set in words: S c (F R c ) (b) Write a set that represents all redhaired male students. 21. In a survey asked of 75 students this morning, 59 said they ate breakfast, 48 said they ate breakfast AND read the Battalion, and 7 said they didn t do either of these things. How many students surveyed read the Battalion?
4 c Dr. Patrice Poage, August 23, A game costs $2.00 to play. This game consists of rolling a 10sided die 1 time. If the die lands on a 9, you win $5. If it lands on a 1 or 8, you win $3. If it lands on a 2, 3, or 4, you win $1. For any other result, you lose. Let X denote the net winnings of a person playing this game. (a) Find the probability distribution for X. (b) How much can a person expect to win/lose if they play this game? Explain. 23. Let n(u) = 26, n(a B) c = 6, n(a c ) = 14, n(b) = 11, find n(a). 24. If you were to shade the Venn diagram to the right to represent the sets below, which numbers would you shade? (a) (A B) C c (b) (B C c ) A A B U 6 2 C 25. A sample of three apples taken from Cavallero s Fruit Stand are examined to determine whether they are good or rotten. (Let g stand for good apple and r stand for rotten apple ) (a) What is an appropriate sample space for this experiment? (b) Describe the event, E, that exactly one of the apples picked is rotten. (c) What is the probability that the last apple picked is rotten? 26. Let U = {1, 2, 5, 7, 9, 10, 11, 12, 14}, A = {1, 5, 9, 12}, B = {2, 7, 12, 14}, C = {1, 10, 11}, and D = {5, 9}. (a) TRUE or FALSE A A (b) TRUE or FALSE (A D) (c) TRUE or FALSE D (A B) (d) TRUE or FALSE n(a c B) = 4 (e) TRUE or FALSE A and B are disjoint. (f) TRUE or FALSE D D c = 27. Shade the venn diagram representing the following set operations: (a) A B c (b) (A B) c (c) (A B) c C (d) A c (B C) c 28. A survey was conducted of 475 students living in dorms at Texas A&M. There were 327 students who said they had a refrigerator in their dorm room, 186 said they had a microwave in their dorm room, and 30 who said they did not have a refrigerator nor a microwave in their dorm room. What is the probability a student selected at random had BOTH a refrigerator and a microwave in their dorm room?
5 c Dr. Patrice Poage, August 23, A foursided die (# d 14) and a fivesided die (# d 15) are rolled at the same time. Let the random variable, X represent the sum of the two dice. Find the probability distribution for this experiment. 30. An experiment consists of rolling a 9sided die (numbered 19) and observing the number on top. Let E be the event that an odd number is rolled. Let F be the event that the number is greater than 5. Are the events E and F mutually exclusive? Mathematically explain why or why not. 31. At a Halloween party some of the guests brought cookies, drinks, or candy. Fill in the Venn Diagram below according to the following information. 5 guest brought cookies, candy and drinks 19 guests brought exactly 2 items 21 guests brought drinks 20 guests did not bring cookies Cookies Drinks U 8 guests brought only candy 2 guests did not bring a drink or candy 5 guests brought only cookies and drinks 11 guests brought drinks and candy Candy 32. In the Venn diagram above, how many people (a) total people were invited? (b) did not bring anything? (c) Brought Cookie or candy, but not drinks? 33. A survey was conducted among 100 college students. Seventyseven said they had a Facebook account. Fiftyfive said they had a MySpace account. Twelve said they did not have an account for either of these. How many students surveyed said they ONLY had a Facebook account? 34. Shade the venn diagrams below to represent: (a) (A B c ) C (b) A (B C) c
6 c Dr. Patrice Poage, August 23, Let U, be a universal set with subsets A, B, and C. CLEARLY circle True or False for each of the following. TRUE FALSE If n(a B) = 0, then A = and B = TRUE FALSE If n(a B) = 0, then A = and B = TRUE FALSE If A B and B C, then A C. 36. A box of tshirts has 7 small, 3 medium, 5 large, and 4 Xlarge tshirts inside. Steven pulls out tshirts 1 at a time until he has 2 small. Let the random variable, X, denote the number of shirts he pulls out of the box. What type of random variable is X? What values may X assume? 37. Let U = {1, 2, 3, 4, 5, 10, 15, 20, 25} A = {4, 10, 20} B = {1, 3, 20, 25} C = {2, 4, 20} CLEARLY CIRCLE either TRUE or FALSE for each of the statements below. (a) TRUE FALSE B B (b) TRUE FALSE n(c B c ) = 5 (c) TRUE FALSE A c (B C) c = (d) TRUE FALSE {4, 10} A (e) TRUE FALSE C 38. A raffle is held in which 2000 tickets are sold at $20 each. There will be one 1st place prize of $10,000, three 2nd place prizes of $5,000, and five 3rd place prizes of $1,000. Let X represent the net winnings for this raffle. (a) Find the probability distribution (reduced fractions or 4 decimal places) (b) How much money could one expect to win or lose if buying a raffle ticket? Explain your answer. 39. Onehundred people were asked what they ate the last time they went to a fast food establishment. Completely fill in the venn diagram according to the results below. Note: two of the sections have been filled in for you 21 did not have any of these items 2 had only a milkshake 47 had french fries 10 had a hamburger and milkshake, but not fries 38 had french fries, but not a milkshake 48 had a hamburger 23 had only french fries Milkshake 2 Hamburger French Fries 21 U
7 c Dr. Patrice Poage, August 23, Over a number of years the grade distribution in a mathematics course was observed (results below). Let X denote the letter grade made, find the probability distribution for X. What is the probably of someone passing this class (making a C or higher)? A B C D F An inspector selects 10 transistors from the production line and notes how many are defective. Determine the event, F, that at most 4 are defective. 42. A survey of 85 shoppers reveals that in the past week, 25 bought poptarts, 41 bought cereal, and 19 bought both poptarts and cereal. How many of the shoppers bought EXACTLY one of these two items? 43. Let A = {1, 2, 3, 4, 5} and B = {5, 10, 15}. Circle either TRUE or FALSE for each of the following statements below. (a) TRUE FALSE {2, 1, 5, 3, 4} A (b) TRUE FALSE {10, 15} B (c) TRUE FALSE A (d) TRUE FALSE 3 A (e) TRUE FALSE B (f) TRUE FALSE Sets A and B are disjoint. 44. Let E and F be two events that are mutually exclusive and P (E) =.37 and P (F c ) =.45. What is the P (E c F c )? 45. An experiment consists of rolling two fair 6sided dice at the same time, and recording the uppermost numbers. What is the probability a sum of 3 is rolled, or one of the dice shows a 2? 46. Let E and F be two events in the same universe. If P (E) = 0.47, P (F ) = 0.18, and P (E F ) = 0.15, find P (E c F ) 47. An experiment consists of rolling a 10 sided die (# d 110) one time. Let E denote the event that the number landing uppermost was even and let F be the event that the number landing uppermost is greater than 4. Are E and F mutually exclusive events? Why or why not? 48. In a survey of all the phones in a store, 50 of the phones had answering machines, 30 of the phones had caller ID, and 70 had an answering machine or caller ID. How many phones had ONLY caller ID? 49. CLEARLY circle which type of random variable each of the following are NOTE: FIN DIS = Finite Discrete, INF DIS = Infinite Discrete, and CONT = Continuous FIN. DIS. INF. DIS. CONT. Number of squirrels in a tree. FIN. DIS. INF. DIS. CONT. The weight of the person sitting next to you. FIN. DIS. INF. DIS. CONT. The volume of water in Lake Bryan. FIN. DIS. INF. DIS. CONT. The # of times you take a class until you pass.
8 c Dr. Patrice Poage, August 23, Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9} with subsets A = {x x is divisible by 3} B = {x x is an odd number} C = {1, 2, 3, 4} D = {1, 5} answer the following question based upon these sets. TRUE FALSE A c C = {1, 2, 4} TRUE FALSE { } = TRUE FALSE A B = {1, 3, 5, 6, 7, 9} TRUE FALSE D has a total of 8 subsets. TRUE FALSE D B TRUE FALSE {2, 4} C 51. A pair of fair dice is rolled, what is the probability that a sum of seven has been tossed if it is known that at least one of the numbers is a 3? 52. Ninetyfive people were surveyed to see which characters they liked from the TV show FRIENDS. Based upon the following results, how many people said they liked ONLY liked Ross? 42 people did NOT like Ross 38 people liked Joey and Ross 71 people liked Monica 10 people like Monica and Ross, but not Joey 16 people liked ONLY Monica 21 people liked Joey, Monica, and Ross 1 person did not like any of them 53. An experiment consists of rolling two 6sided fair dice. What is the probability the sum of the numbers is less than nine, if it is known that both of the dice rolled a number greater than two? 54. Larry goes to a charity event. He is given the opportunity to purchase a raffle ticket for a $75 door prize. The cost of the ticket is $3, and 250 tickets will be sold. Determine Larry s expected net win/loss if he purchases one ticket. 55. Let E and F be sets in sample space, S, with P (E) = 0.49, P (F ) = 0.44, and P (E F ) = Find P (E c F c ). 56. A survey was conducted of 120 students on whether or not they read the last two books in the Twilight series: Eclipse and Breaking Dawn. Nineteen students had read neither book, while 89 students had read Breaking Dawn and 96 students had read Eclipse. What is the probability a student read both books? 57. Two fair 6sided dice (# d 16) are rolled at the same time. One of the dice is blue and the other is red. What is the probability the sum is greater than 5, if it is known that the blue die rolled a number great than 2?
9 c Dr. Patrice Poage, August 23, A drawer contains 4 blackink pens, 3 blueink pens, and 3 redink pens. You randomly reach in and select one pen out at a time (without replacement) until you have 2 blackink pens. Let X be the random variable denoting the total number of pens you draw out. What values may X assume? 59. A survey was taken among 77 males and 73 females to find out their favorite sport to watch on TV. Based on the results in the table below, if a person from this group was randomly selected, find (a) the probability the person is a male, given they like football best. (b) the probability the person is a female and likes basketball best. (c) the probability the person likes baseball best, given they are a male. (d) The odds in favor of a female choosing baseball. (e) The odds against a person liking football best, given they are a female. 60. Shade the venn diagrams below to represent: (a) (A B C) c (b) (A B C) c 61. Shade the venn diagrams below to represent: (a) (A c B) C (b) A (B c C) 62. Decide if each of the random variables are Finite Discrete, B) Infinite Discrete, or C) Continuous AND underneath each, WRITE OUT THE VALUES X may assume. Let X be the number of times a die is thrown until a 2 appears. Let X be the number of broken candy canes in a pack of 16. Let the random variable X represent the number of hours you exercised in the last week. A bowl has 4 blue M&M s and 8 red M&M s in it. An M&M is chosen, without replacement, until a blue M&M is picked. Let X be the number of picks needed to get a blue M&M. 63. A raffle offers a first prize of $400 and 5 second prizes of $80 each. One ticket costs $3, and 500 tickets are sold. Let X denote the net winnings. Find the expected value of X.
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