1

Math 102 Final Exam Fall 2017

Name: _________ ______________ _ Please write your name clearly.

Directions: Provide complete solutions to each problem in the space provided on these pages. Use the backs of these pages should additional space be required. Credit will be awarded for fully supported solutions. Therefore, please show all of your work for full credit. Remember, academic integrity is required at all times. ______________________________________________________________________________________________________________________ 1. If set 𝐴 = {1, 3, 5, 7, 9, 11} a. Describe set A using a set-builder notation.

b. How many proper subsets does set A has?

2. Let 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 𝐴 = {1, 2, 4, 7, 9} 𝐵 = {3, 4, 6, 9, 10} 𝐶 = {1, 4, 5, 6, 8}

a. Find 𝐴 ∩ 𝐶′

b. Find (𝐴 ∩ 𝐵) ∪ (𝐵 ∩ 𝐶) ∪ (𝐴 ∩ 𝐶)

2

3. Use the Venn diagram to illustrate the situation below and then Find: 𝐶 − (𝐴′ ∩ 𝐵′) 𝑼 = {𝒂, 𝒃, 𝒄, 𝒅, 𝒆, 𝒇, 𝒈, 𝒉, 𝒊, 𝒋}

𝐴 = {𝑏, 𝑐, 𝑒, 𝑖} 𝐵 = {𝑎, 𝑒, 𝑔, ℎ, 𝑖} 𝐶 = {𝑏, 𝑓, ℎ, 𝑖, 𝑗}

4. A survey of 80 kids about their favorite ice-cream flavor shows that:

40 Kids like Strawberry.

42 Kids like Chocolate.

50 Kids like Vanilla.

17 Kids like Chocolate and Strawberry.

21 Kids like Vanilla and Strawberry.

24 Kids like Vanilla and Chocolate.

9 Kids like Vanilla, Chocolate and Strawberry.

a. How many kids like exactly 2 ice-cream flavors?

b. How many kids like exactly 1 ice-cream flavor?

c. How many kids do not like any of the 3 ice-cream flavors?

d. How many kids like Vanilla and Chocolate but not Strawberry?

e. How many kids like Chocolate and Strawberry but not Vanilla?

3

5. Negate the following: a. Some courses are easy.

b. All athletes are healthy.

6. Consider the following statements.

g: I will go to college.

j: I will join the army.

a. Write the conjunction of the given statements in both symbols and words.

b. Write the disjunction of the given statements in both symbols and words.

c. Write the negation of the conjunction of the given statements in both symbols and words.

d. Write the negation of the disjunction of the given statements in both symbols and words.

7. Use truth tables to show that: ~𝑝 → 𝑞 ⟺ (𝑝 ∨ 𝑞)

4

8. Given the conditional statement: “If I wake up early, then I will not miss the test.” a. Write the converse of the given conditional statement.

b. Write the inverse of the given conditional statement.

c. Write the contrapositive of the given conditional statement.

9. Use Euler diagrams to validate the following argument: All actors are beautiful.

Ms. Brown is beautiful. ∴ Ms. Brown is an actor.

10. Use truth table to determine the validity of the following argument: 𝑝 ∨ ~𝑞

𝑝 ∴ 𝑞

5

11. In Connecticut, auto license plates carry 3 digits followed by 3 letters. The digits can be any integer from 0 through 9, and the letters can be any letter from the 26 English alphabets. How many different license plates can be formed if the letters can repeat in the license plate but integers cannot?

12. Compute: 𝑃(7, 3) ÷ 3!

13. If you have 6 coins, a penny, nickel, dime, quarter, half a dollar and a silver dollar. How many different sums of money you can make using 3 coins.

6

14. In how many ways can you arrange the letters in the word “success”?

15. Suppose that three fair coins are tossed in the same time. Find the probability that 2 heads and 1 tails will come up. Hint: A tree diagram might be helpful.

16. Tim and Tom both belong to a club of 25 members. A committee of 5 is to be selected at random from the 25 members. Find the probability that both Tim and Tom will be selected.

7

17. A fan is selected at random from the NBA final game. The probability that the fan likes Cleveland only is 0.43. The probability the fan likes both Cleveland and Golden State is 0.03. The probability that the fan likes Cleveland or Golden State is 0.87. What is the probability that the fan likes Golden State only?

18. Walmart notices that during Super Bowl weekend, shoppers often buy related foods. The chance that people will buy beer is 0.5, and the probability that they will buy both beer and chips is 0.2. What is the probability that a shopper will buy chips, given that they also bought beer?

8

19. A company has introduced 3 new products to the market A, B and C. The probability of success for product A is 0.82, and the probability of success for product B is 0.67, and the probability of success for product C is 0.73. Find the probability that exactly 2 products will be successful.

20. The probability that Jessica will win the swimming competition is 19

32 .What are the odds in favor

of her losing the swimming competition?

9

21. Below are the grades of 16 students in a final exam. 73 52 78 66 67 71 69 78

61 77 92 88 94 57 75 69

Use the groups: 50 – 59, 60 – 69, 70 – 79, 80 - 89 and 90 – 99 to answer questions a and b. a. Create a frequency distribution for the given grades.

b. Create the corresponding histogram.

10

Use the following information to answer questions 22 and 23. 22. The data below are the numbers of cigarettes Terri has smoked each week for the last 5 weeks. 79 83 94 79 65

a. Find the mean of the given data.

b. Find the median of the given data.

c. Find the mode of the given data.

d. Find the range of the given data.

23. Calculate the standard deviation of the given data.

11

24. 350 students with normally distributed salaries earn an average of $6,800 with a standard deviation of $1,800.

a. How many student’s income is less than one standard deviation above the main?

Hint: The normal distribution curve provided above might be helpful.

b. If a student is selected randomly, what is the probability that the student’s income is less than $7,250? Hint: You will need to refer to table II on the last page of this final exam.

12

25. 500 students are surveyed about their favorite subject. The result of the survey is shown in the table below.

English (E) Math (M) History (H) Physics (P) Chemistry (C)

175 50 100 75 100

a. Present the given data in a line graph.

b. Present the given data in a bar graph.

c. Present the given data in a circle graph.

13