MTH 103 H Final Exam. 1. I study and I pass the course is an example of a. (a) conjunction (b) disjunction. (c) conditional (d) connective

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1 MTH 103 H Final Exam Name: 1. I study and I pass the course is an example of a (a) conjunction (b) disjunction (c) conditional (d) connective 2. Which of the following is equivalent to (p q)? (a) p q (b) p q (c) q p (d) p q A merchant surveyed 400 people to determine the way they learned about tan upcoming sale. The survey showed that 200 learned about the sale from the radio, 180 from television, 150 from the newspaper, 80 from radio and television, 90 from radio and newspapers, 50 from television and newspaper, and 30 from all three sources. Use this information to answer the next few problems. 3. How many people learned of the sale from either the newspaper and not the television or the television and not the newspaper? (a) 50 (b) 120 (c) 230 (d) How many of the people learned of the sale only from the radio? (a) 30 (b) 60 (c) 100 (d) How many people had not learned of the sale? (a) 0 (b) 60 (c) 70 (d) 180 1

2 6. Which of the following is not a binomial random variable? (a) A die is tossed eight times and the number of times a four comes up is counted. (b) A die is tossed five times and the sum of the numbers that occur is observed. (c) Fifteen women are surveyed and asked whether their car is blue or not. (d) A class is split up into three teams - A, B, and C. Each week for four weeks a team is selected at random. The number of times team C is selected is observed. 7. A day of the week is picked at random. What are the odds that the day is a Tuesday? (a) 1 to 7 (b) 7 to 1 (c) 1 to 6 (d) 1 to 8 8. Which of the following is not a subset of the set {1, 2, 3, 4, 5} (a) (b) {1} (c) {1, 2} {2, 3} (d) {1, 2, 3, 4, 5} (e) They are all subsets 9. How many subsets does the set {1, 2, 3, 4, 5} have? (a) 5 (b) 10 (c) 25 (d) I have four books and five dvd s to put on a bookshelf. If I want all the books together and all the dvd s together, how many ways can this be done? (a) 20 (b) 9! (c) 2 4! 5! (d)

3 11. If P r(e F ) = 0.4, P r(f E) = 0.5, and E and F are independent events. Find P r(e F ) (a) 0.2 (b) 0.7 (c) 0.8 (d) Let E 1 be the event Transistor 1 doesn t fail within the first 1000 hours. Let E 2 be the event Transistor 2 doesn t fail within the first 1000 hours. Let E 3 be defined similarly. Transistor 1, 2, and 3 are put into a radio and the radio fails if and only if all three transistors fail. Which of the following will give the probability the radio stops working in the first 1000 hours? (a) P r(e 1 E 2 E 3) (b) P r(e 1 E 2 E 3) (c) P r(e 1 E 2 E 3 ) (d) P r((e 1 E 2 E 3 ) ) 13. Which of the following pairs of events are mutually exclusive? (a) Two cards are drawn from a deck without replacement. E = The first card is a heart and F = The second card is black (b) An experiment has sample space S = {a, b, c, d, e}. E = {a, b, c} and F = {c, d, e} (c) A coin is flipped 5 times. E = The first flip is a head and F = All tales occur. (d) A die is tossed twice. E = The first roll is a 2 and F = The sum of the two rolls is odd. 14. How many three digit area codes can there be if the first number has to be odd and the last number is a 0 or 1? (a) 100 (b) 120 (c) 500 (d) An archer has probability 0.3 of hitting a certain target. What is the probability of hitting the target exactly two times in five attempts? (a) 20(0.3) 2 (0.7) 3 (b) 10(0.3) 3 (0.7) 2 (c) 10(0.3) 2 (0.7) 3 (d) 10(0.3) 2 (0.7) 5 3

4 16. State the converse and contrapositive of the statement If my memory serves me correctly, I was nine years old when I moved. (a) converse: (b) contrapositive: (c) Which is equivalent to the original statement? 17. Construct a truth table for the following and determine whether it is a tautology, contradiction, or neither. (p q) r r 18. Show ( q r) p ( q p) ( r p) 4

5 19. Let the universal set be the set of all letters of the English alphabet. Let p(x) be the statement x is a vowel. Let q(x) be the statement x is a consonant. Determine the truth value of each of the following. (a) x[p(x) q(x)] (b) x[p(x) q(x)] (c) x[p(x) q(x)] (d) [ xp(x)] [ xq(x)] 20. Negate the following statements by changing existential quantifiers to universal quantifiers, or vice versa. (a) Some students show up on time. (b) None of my cousins are tall. (c) He goes out every night. (d) There is a planet that takes 300 years to orbit the sun. 21. A certain organization has 9 members, 5 of which are women. In how many ways can a committee consisting of two men and two women be formed? 5

6 22. Show that the following argument is valid If I have money, then I don t stay home. I stay home or I talk to my friends. I don t talk to my friends. Therefore I don t have money. 23. U = {a, b, c, d, e, f} R = {a, b, d, f} S = {a, c, f} T = {b, c} (a) R S T (b) R S T (c) R (S T ) (d) S T (e) R R 6

7 24. In a certain carnival game the player selects two balls at random from an urn containing two red balls and four white balls. The player receives $5 if he draws two red balls and $1 if he draws one red ball. He loses $1 if neither ball is red. (a) Determine the probability distribution for the experiment of playing the game and observing the player s earnings. (b) What are the player s expected earnings? (c) Given your answer from part (b) would it be wise for the player to continue playing the game many times? 25. Expand (x + y) 7 using the binomial theorem 7

8 26. An exam consists of ten multiple-choice questions where each question has four choices. (a) If you guess the answers completely at random, what is the probability that you answer at least two questions correctly? (b) If every student in the class were to guess at random, what score on average would they be expected to receive? 27. Lucy and Ethel play a game of chance in which a pair of fair dice is rolled once. If the result is 7 or 11, then Lucy pays Ethel $10. Otherwise, Ethel pays Lucy $3. Calculate Ethel s expected earnings. 8

9 28. Twenty percent of the library books in the fiction section are worn and need replacement. Ten percent of the nonfiction holdings are worn and need replacement. The library s holdings are 40% fiction and 60% nonfiction. (a) Draw a tree diagram illustrating the situation. (b) Find the probability that a book chosen at random from this library is worn and needs replacement. (c) Given that a randomly chosen book from this library is worn and needs replacement, what is the probability that this book is nonfiction? 29. Assume that a certain school contains an equal number of female and male students and that 5% of the male population is color-blind. Find the probability that a randomly selected student is a color-blind male. 9

10 Bonus k Pr (X=k) Given the random variable X has the probability distribution as above, find the probability distribution for the random variable (X 1) 2 2. How much did you enjoy this course? (a) Not at all. Time would have better been spent sleeping. (b) I learned some interesting things. (c) Best course ever... wish I had majored in math! (d) 10