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MATH 166 Exam I Sample Questions PART I-MULTIPLE CHOICE 1. Let p be the statement Pi is a rational number. (which is false) and q √ be the statement 4 is a rational number (which is true). Which of the following is true? (a) p∧q (b) ∼ p∨q (c) p∨ ∼ q (d) p∧ ∼ q (e) None of these are true 2. Given the Venn Diagram below, the set (a) b (b) a, b, c (c) a, b, c, h (A ∪ B) ∩ C c consists of which region(s)? only only only (d) None of these (e) a, b, c, e, h only U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 3, 5, 7}, A ∪ Bc? 3. Let 4. (a) {2} (b) {0, 4, 6, 8} (c) {0, 2, 4, 6, 8} (d) {0, 1, 4, 6, 8, 9} (e) {0, 2, 3, 4, 5, 6, 7, 8} Using the sets in #3 above {2} ⊂ A. , let p p is true; q is true. (b) p is true; q is false. (c) p is false; q is true. (d) p is false; q is false. (e) None of these B = {1, 3, 5, 7, 9}. be the statement {3, Which of the following is correct? (a) and Which of the following is 5} ∈ A ∩ B and q be the statement 5. In the SEC (14 total schools), 8 schools have winning records in Women's Volleyball, 11 schools have winning records in Women's Soccer, and 2 schools do not have winning records in either sport. How many schools have winning records in both sports? (a) 6 (b) 7 (c) None of these (d) 9 (e) 17 6. Two fair 4-sided dice are rolled. What is the probability that the absolute value of the dierence between the rst and second die is at least one? (For example, if the dice are (1, 3), then (a) (b) (c) (d) (e) |1 − 3| = 2). 3 5 5 6 2 7 3 4 3 8 7. Two fair 4-sided dice are rolled. What is the probability that at least one of the numbers is a 3? (a) (b) (c) (d) (e) 7 16 1 8 1 2 3 8 3 10 8. According to the show Guys and Dolls, if a guy is acting strange, It's a probable 12 to 7 that the guy's only doing it for some doll. If these are the odds the man is in love, what is the probability that the man is in love? (a) (b) (c) (d) (e) 7 12 7 19 5 12 12 19 5 19 9. In a u study of 400 individuals, 145 of them contracted the u. Of those who contracted the u, 60 had been vaccinated, while 190 of those who did not contract the u were vaccinated. If an individual from the study is chosen at random, what is the probability they were vaccinated given they did not contract the u? (a) 190 400 (b) None of these (c) (d) (e) 190 255 255 400 250 400 10. Given the following: P (A) = 0.2, P (B) = 0.6, P (Ac ∩ B c ) = 0.2 P (C) = 0.5, P (D) = 0.3, P (C c ∩ Dc ) = 0.35 Which statement is correct? (a) (b) (c) (d) (e) A A A A A and and and and and B B B B B are mutually exclusive, neither, C and D C and are neither, C and D are independent are independent D are neither C and D are neither C and D are mutually exclusive are mutually exclusive, are independent, 11. Use the tree diagram below to nd P (C): (a) 0.42 (b) 0.9 (c) 0.18 (d) 0.45 (e) None of these 12. Use the tree diagram in #11 to nd (a) (b) (c) 3 5 3 7 21 50 (d) None of these (e) 3 10 P (F |C): U be the set of all MATH 166 students. Let B = {x|x is a business and F = {x|x is a freshman}. Which of the following describes the set of 13. Let major}, M = {x|x is male}, all MATH 166 students who are male and either not business majors or not freshman or not either one? (a) (M ∩ B c ) ∪ F c (b) (M ∪ B c ) ∩ F c (c) M ∩ (B c ∪ F c ) (d) M ∪ (B c ∩ F c ) (e) More than one of these is correct. 14. Let p: q: r: p, q, r be the following statements: I was in the Corps of Cadets I attended a Bonre cut I attended all Aggie home football games (a) Write the statement ∼ p∧ ∼ q ∧ r in words. (b) Write the symbolic form of I was in the Corps of Cadets and I have either not attended a Bonre cut or not attended all the Aggie home football games or both. 15. (a) (8 pts) Construct a truth table for (∼ p ∧ q) ∧ (∼ p ∨ q). (NOTE: Not all columns of the table have to be used) p q T T F F T F T F (b) Is (∼ p ∧ q) ∧ (∼ p ∨ q) a contradiction, tautology, or neither? 16. In a campus survey regarding news sources: 4 students use a newspaper, TV, and the Internet for news 20 students use exactly 2 sources for news 14 students use a newspaper for news 43 students do NOT use the Internet for news 13 students use only TV for news 50 students do NOT use TV or a newspaper for news 2 students use only the newspaper and the Internet for news 9 students use TV and a newspaper for news Completely ll in the diagram below and determine how many students participated in the survey (where use a newspaper}, and N = {students who I = {students who use the Internet}). T = {students who use TV}, 17. The numbers 1, 2, and 10 are each written on a card and placed in a hat. Two numbers are drawn at the same time, and the product of the numbers is recorded. List the sample space for this experiment. Are the outcomes equally likely? S = {s1 , s2 , s3 , s4 , s5 } P (s3 ) = 0.2. 18. Let (a) If s4 (b) If A = {s1 , s3 , s5 } be the sample space of an experiment with is twice as likely to occur as and s5 , B = {s3 , s4 }, P (s1 ) = 0.2, P (s2 ) = 0.15, and what are their probabilities? what is P (A ∩ B c )? 19. According to the internet (which must be true, right?), 90% of all email received is spam. Suppose your spam lter program will correctly mark spam email 95% of the time, but also mark legitimate email as spam 3% of the time. Given a message is not marked as spam, what is the probability that it really is not spam?