Exam 2 Review F09 O Brien. Finite Mathematics Exam 2 Review


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1 Finite Mathematics Exam Review Approximately 5 0% of the questions on Exam will come from Chapters, 4, and 5. The remaining 70 75% will come from Chapter 7. To help you prepare for the first part of the exam, I would suggest you rework Exam. If you have difficulty with a particular problem, work additional problems of that type from the Exam Review and/or the homework assignments. You do not have to complete this review sequentially. Take a sample of problems from different sections and work them like a practice exam. Read and carefully follow all directions. Show all of your work on every problem. Work the problems on blank paper with no resources other than the departmental formula sheet and your calculator. If you have to peek at something (homework, notes, textbook, solutions manual, etc.) to complete a problem, take note of what knowledge you were missing, and work some additional problems of this type.. Let A {a, b, c, d, e}, B {c, d, e}, C {f, g}, and U {a, b, c, d, e, f, g, h}. 7. Indicate whether each of the following four statements is true or false. a. g B b. c C c. A B d. C B e. How many subsets does set A have? f. Which two sets are disjoint?. Let A {,, }, B {,, 5}, C {6, 7}, and U {0,,,, 4, 5, 6, 7}. List the members of each of the following sets using set braces. 7. a. B' b. A U C c. A' B d. C' (A' U B). Let U {all students at NWACC}, F {all students taking Finite Math}, A {all students taking accounting}, B {all students taking Business Foundations} 7. Describe each of the following sets in words: a. (F B)' b. F' U B' c. A U (B F') 4. Shade in the part of a Venn diagram representing: 7. a. (A U B') C b. A (B U C') c. A' (B C)' d. (A' B') C 5. Use the union rule to answer the following questions. 7. a. If n(a) 5, n(b) 8, and n(a B) 4, what is n(a U B)? b. If n(a), n(b) 7, and n(a U B) 0, what is n(a B)? c. If n(b) 7, n(a B) and n(a U B) 0, what is n(a)? 6. Draw a Venn diagram and use the given information to fill in the number of elements for each region. a. n(a'), n(b) 5, n(a' U B') 46, n (A B) 7. b. n(a) 54, n(a B), n(a U B) 85, n(a B C) 4, n(a C) 5, n(b C) 6, n(c) 44, n(b') 6 7. Show that the given statement is true by drawing Venn diagrams and shading the regions representing the sets on each side of the equal sign. 7. a. (A B)' A' U B' b. A U (B C) (A U B) (A U C)
2 8. A researcher collecting data on 00 households finds that: 7. have a DVD player; 56 have a VCR, have both. Use this information to create a Venn diagram to answer the following questions. a. How many do not have a VCR? b. How many have neither a DVD player nor a VCR? c. How many have a DVD player but not a VCR? business students were surveyed and the following information was collected: 7. 9 read Business Week (BW); 8 read the Wall Street Journal (WSJ); 50 read Fortune magazine (F); read Business Week and the Wall Street Journal; read the Wall Street Journal and Fortune; read Business Week and Fortune; 9 read all three Use this information to create a Venn diagram to answer the following questions. a. How many students read none of the publications? b. How many read only Fortune? c. How many read Business Week and the Wall Street Journal, but not Fortune? 0. A telephone survey of television viewers revealed the following information: 7. 0 watch situation comedies; 9 watch game shows; 7 watch movies; watch all three; 9 watch movies but not game shows; 5 watch situation comedies but not game shows; 0 watch both situation comedies and movies; 7 watch none of these Use this information to create a Venn diagram to answer the following questions. a. How many viewers were interviewed? b. How many viewers watch comedies and movies but not game shows? c. How many viewers watch only movies? d. How many viewers do not watch movies?. Write the sample spaces for the following experiments. 7. a. a coin is tossed and a single die is rolled b. a box contains four balls, numbered,,, and 4. One ball is drawn at random, its number is recorded and the ball is set aside. The box is shaken and a second ball is drawn and its number is recorded.. For the following experiments, write the sample space, give the value of n(s), tell whether the outcomes in S are equally likely, and then write the indicated events (E) in set notation. 7. I. A committee of three is to be selected from 5 individual: Alan, Ben, Cara, Debbie, and Ellen. a. Cara is on the committee. b. Debbie and Ellen are not both on the committee. c. Both Alan and Ben are on the committee.
3 II. An unprepared student takes a threequestion true/false quiz in which he guesses the answers to all three questions, so each answer is equally likely to be correct or incorrect. a. The student gets three wrong answers. b. The student gets exactly two answers correct. c. The student gets only the first answer correct.. A single fair die is rolled. Find the following probabilities. 7. a. Getting a number less than 5 b. Getting an odd number 4. A survey of people attending a Lunar New Year celebration yielded the following results: 7., 7. 0 were women; 50 spoke Cantonese; 70 lit firecrackers; 08 of the men spoke Cantonese; 00 of the men did not light firecrackers; 8 of the noncantonesespeaking women lit firecrackers; 78 of the noncantonesespeaking men did not light firecrackers; 0 of the women who spoke Cantonese lit firecrackers. Find the probability that a randomly chosen attendee: a. did not speak Cantonese. b. was a woman who did not light a firecracker. c. was a Cantonesespeaking man who lit a firecracker. 5. A card is drawn from a wellshuffled deck of 5 cards. Find the probabilities of drawing the. 7., 7.4 a. a heart or a black card b. a queen or a spade c. a diamond or a face card 6. A pair of standard dice is to be rolled. Find the probability that: 7.4 a. a sum less that 5 is rolled b. a sum less than or greater than 8 is rolled c. the second die is a 5 or a sum of 0 is rolled 7. Given P(E).4, P(F).5, and P(E U F).59, use a Venn diagram to find the following. 7.4 a. P(E' F') b. P(E' U F') c. P(E' U F) d. P(E F') 8. A marble is drawn from a box containing red, 6 yellow, and 5 green marbles. Find the following. 7.4 a. odds in favor of drawing a yellow marble b. odds against drawing a green marble 9. Two fair dice are rolled. Find the odds of rolling a 5 or a A fastfood chain is conducting a game in which the odds of winning a double cheeseburger 7.4 are reported to be to 00. Find the probability of winning a double cheeseburger.. If two fair dice are rolled, find the probabilities of the following results. 7.5 a. sum of 8, given that the sum > 7 b. a sum of 6, given a double was rolled
4 . Two cards are drawn in succession, without replacement, from a deck of What is the probability that: a. the second card was a spade, given that the first card was a spade? b. the second card was an ace, given that the first card was a face card? c. the second card was a club, given that the first card was a heart or a spade? d. the second card was a nine, given that the first was a diamond and a nine?. A new drug was administered to 00 people. Six of the people reported a rise in blood pressure, reported loss of sleep, and reported both of these side effects. Find the probability that a person taking the drug will experience: a. a rise in blood pressure, given a report of lost sleep b. loss of sleep, given a reported rise in blood pressure c. neither of these two side effects 4. If A and B are events such that P(A).5 and P(A U B).7, find P(B) when 7.5 a. A and B are mutually exclusive b. A and B are independent 5. The Midtown Bank has found that most customers at the tellers windows either cash a check or 7.5 make a deposit. The following table gives the transactions for one teller for one day. Cash Check No Check Totals Make Deposit No Deposit Totals Letting C cash check and D make deposit, express each probability in words and find its value. a. P(C D) b. P(D' C) c. P(C' D') d. P[(C D)'] 6. In a certain area, 5% of the population are joggers and 40% of the joggers are women. 7.5 If 55% of those who do not jog are women, find the probabilities that a randomly chosen individual from the community fits the following description: a. a woman jogger b. not a jogger c. a woman d. Are the two events, woman and jogger, independent events? 7. A federal study showed that 49% of all those involved in a fatal car crash wore seat belts. 7.6 Of those in a fatal crash that wore seat belts, 44% were injured and 7% were killed. For those not wearing seat belts, 4% were injured and 50% were killed. a. Find the probability that a randomly selected person who was killed in a car crash was wearing a seat belt. b. Find the probability that a randomly selected person who was unharmed in a fatal car crash was not wearing a seat belt. 4
5 8. One box, A, contains three red and four green balls, while a second box, B, contains 8 red 7.6 and green balls. A box is selected at random and a ball is drawn. What is the probability that: a. the ball is red, given that it came from box A? b. the ball is red and came from box B? c. the ball is red? d. the ball was drawn from box B, given that it is red? 9. It is known that about 5% of all men and.5% of all women are color blind. Assume that half 7.6 of the population are men and half women. If a person is selected at random, what is the probability that the person is: a. female and color blind? b. color blind, given that the person is male? c. color blind? d. male, given that is color blind? 0. Records indicate that % of the population has a certain kind of cancer. A medical test has 7.6 been devised to help detect this kind of cancer. If a person does have the cancer, the test will detect it 98% of the time. However, % of the time the test will indicate that a person has the cancer when, in fact, he or she does not. For persons using this test, what is the probability that: a. the person has this type of cancer, and the test indicates that he or she has it? b. the person has this type of cancer, given that the test indicates that he or she has it? c. the person does not have this type of cancer, given a positive test result for it? 5
6 Answers. a. F b. T c. F d. T e. f. A & C, and B & C. a. B' {0,, 4, 6, 7} b. A U C {,,, 6, 7} c. A' B {5} d. C' (A' U B) {0,,, 4, 5} a. (F B)' all students who are not taking Finite Math and Business Foundations b. F' U B' all students who are not taking Finite Math or who are not taking Business Foundations c. A U (B F') all students who are taking accounting or who are taking Business Foundations but not Finite Math 4a. (A U B') C b. A (B U C') c. A' (B C)' d. (A' B') C 5a. n(a U B) 9 b. n(a B) 9 c. n(a) 6 6a. b. 7a. b. (A B)' A' U B' A U (B C) (A U B) (A U C) 6
7 8. a. 44 do not have a VCR b. 5 have neither c. 9 have a DVD player but not a VCR 9. a. Only student reads none of the three publications. b. There are 5 students who read only fortune. c. 4 students read Business Week and the Wall Street Journal, but not Fortune. 0. a. n(u) 5 b. n(c G' M) 7 c. n(c' G' M) d. n(m') 6 a. S {(H, ), (H, ), (H, ), (H, 4), (H, 5), (H, 6), (T, ), (T, ), (T, ), (T, 4), (T, 5), (T, 6)} b. S {(, ), (, ), (, 4), (, ), (, ), (, 4), (, ), (, ), (, 4), (4, ), (4, ), (4, )}. I. S {ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE}; n(s) 0; Assuming the committee members are selected at random, the outcomes are equally likely. a. E {ABC, ACD, ACE, BCD, BCE, CDE} b. E {ABC, ABD, ABE, ACD, ACE, BCD, BCE} c. E {ABC, ABD, ABE} II. S {CCC, CCW, CWC, CWW, WCC, WCW, WWC, WWW}; n(s) 8; outcomes are equally likely a. E {WWW} b. E {CCW, CWC, WCC} c. E {CWW}. a. P(n < 5) 4 b. P(n is odd) a. P(speaks did not speak Cantonese) b. P(woman who did not light firecracker) c. P(Cantonesespeaking male who lit a firecracker)
8 5. a. P(Heart or Black) b. P(Queen or Spade) c. P(Diamond or Face) a. P(sum < 5) 6 b. P(sum < or sum > 8) 6 c. P( nd die is 5 or sum is 0) a. P(E' F').4 b. P(E' U F').8 c. P(E' U F).76 d. P(E F').4 8. a. O(Y) to 4 b. O(G') 9 to 5 9. O (5 or 9) to 7 0. P(win) 0. a. P(sum 8 sum > 7) 5 b. P(sum 6 double) a. P( nd spade st spade) 5 c. P( nd club st heart or spade) 5 b. P( nd ace st 4 face card) 5 d. P( nd nine st nine and diamond) 5 P(R IL).0. a. P(R L).4 P(L).05 5 b. P(L R) P L ( IR) P( R) c. P(L U R)' a. P(B). b. P(B).4 5. a. P(C D) probability a customer cashes a check, given that he/she made a deposit 4 b. P(D' C) probability a customer does not make a deposit, given that he/she cashed a check c. P(C' D') probability a customer does not cash a check, given he/she did not make a deposit 4 d. P[(C D)'] probability a customer does not cash a check and make a deposit 6. a. P(W J).060 b. P(J').85 c. P(W).575 d. No, the two events are not independent. P(W J) P(W) P(J). 8
9 7. a. P(killed wore a seat belt).46 b. P(not wearing seatbelt unharmed) a. P(R A) 7 4 c. P(R) 70 b. P(B R) 5 8 d. P(B R) 4 9. a. P(W I CB).005 b. P(CB M).05 0 c. P(CB).065 d. P(M CB) 0. a. P(C P).096 b. P(C P) 5.4 c. P(C' P) 5.6 9
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