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Topic 1: Determining a sample space and outcomes for an event: Experiment involving multiple selections Problem 1: A coin is flipped twice. List the sample space and the event of getting at least one head. Answer: Sample space = {HH, HT, TH, TT} (4 outcomes). Event (at least one head) = {HH, HT, TH} (3 outcomes). Problem 2: Two cards are drawn from {A, B, C} with replacement. List the sample space and the event of drawing A at least once. Answer: Sample space = {AA, AB, AC, BA, BB, BC, CA, CB, CC} (9 outcomes). Event (A at least once) = {AA, AB, AC, BA, CA} (5 outcomes). Topic 2: Determining outcomes for unions, intersections, and complements of events Problem 1: For a die roll, let A = {1, 2, 3} (numbers ≤ 3) and B = {2, 4, 6} (even numbers). Find A ∪ B, A ∩ B, and A'. Answer: Sample space = {1, 2, 3, 4, 5, 6}. A ∪ B = {1, 2, 3, 4, 6}, A ∩ B = {2}, A' = {4, 5, 6}. Problem 2: For choosing a letter from {A, B, C, D}, let C = {A, B} and D = {B, C}. Find C ∪ D, C ∩ D, and D'. Answer: Sample space = {A, B, C, D}. C ∪ D = {A, B, C}, C ∩ D = {B}, D' = {A, D}. Topic 3: Using a Venn diagram to understand the addition rule for probability Topic 4: Computing probability involving the addition rule using a two-way frequency table Problem 1: A table shows: 10 males like sports, 5 don’t; 6 females like sports, 4 don’t. Find P(likes sports or male). Answer: Total = 25. Likes sports = 10 + 6 = 16, males = 10 + 5 = 15, male and likes sports = 10. P(sports ∪ male) = P(sports) + P(male) - P(sports ∩ male) = 16/25 + 15/25 - 10/25 = 21/25 = 0.84. Problem 2: Table: 12 adults prefer tea, 8 coffee; 10 kids prefer juice, 5 soda. Find P(prefers tea or is a kid). Answer: Total = 35. Prefers tea = 12, kids = 15, kids and tea = 0 (assumed). P(tea ∪ kid) = 12/35 + 15/35 - 0/35 = 27/35 ≈ 0.771. Topic 5: Calculating relative frequencies in a contingency table Problem 1: Table: 8 males pass, 2 fail; 6 females pass, 4 fail. Find the relative frequency of passing among males. Answer: Total males = 8 + 2 = 10. Relative frequency = 8/10 = 0.8. Problem 2: Table: 15 adults like tea, 5 coffee; 10 kids like juice, 3 soda. Find relative frequency of kids liking juice. Answer: Total kids = 10 + 3 = 13. Relative frequency = 10/13 ≈ 0.769. Topic 6: Computing conditional probability using a sample space Problem 1: A die is rolled. Sample space = {1, 2, 3, 4, 5, 6}. Find P(even | number > 3). Answer: Event (number > 3) = {4, 5, 6} (3 outcomes). Even and > 3 = {4, 6} (2 outcomes). P(even | > 3) = 2/3 ≈ 0.667. Problem 2: Choose a letter from {A, B, C, D}. Find P(vowel | letter in {A, B}). Assume A is a vowel. Answer: Event (letter in {A, B}) = {A, B} (2 outcomes). Vowel and in {A, B} = {A} (1 outcome). P(vowel | {A, B}) = 1/2 = 0.5. Topic 7: Computing conditional probability using a two-way frequency table Problem 1: Table: 10 males like sports, 5 don’t; 6 females like sports, 4 don’t. Find P(likes sports | male). Answer: Total males = 15. Males who like sports = 10. P(likes sports | male) = 10/15 = 2/3 ≈ 0.667. Problem 2: Table: 12 adults prefer tea, 8 coffee; 10 kids prefer juice, 5 soda. Find P(prefers juice | kid). Answer: Total kids = 15. Kids who prefer juice = 10. P(juice | kid) = 10/15 = 2/3 ≈ 0.667. Topic 8: Computing conditional probability using a large two-way frequency table Problem 1: Table: 50 males pass math, 20 fail; 30 females pass, 10 fail. Find P(passes math | female). Answer: Total females = 30 + 10 = 40. Females who pass = 30. P(passes | female) = 30/40 = 3/4 = 0.75. Problem 2: Table: 100 adults like tea, 50 coffee, 25 juice; 80 kids like juice, 20 soda, 10 tea. Find P(likes tea | adult). Answer: Total adults = 100 + 50 + 25 = 175. Adults who like tea = 100. P(tea | adult) = 100/175 = 4/7 ≈ 0.571. Topic 9: Computing conditional probability to make an inference using a two-way frequency table Problem 1: Table: 12 males vote yes, 8 no; 10 females vote yes, 5 no. Infer if males are more likely to vote yes than females. Answer: P(yes | male) = 12/20 = 0.6. P(yes | female) = 10/15 ≈ 0.667. Inference: Females are slightly more likely to vote yes (0.667 > 0.6). Problem 2: Table: 20 adults exercise daily, 10 weekly; 15 kids exercise daily, 5 weekly. Infer if adults are more likely to exercise daily. Answer: P(daily | adult) = 20/30 = 2/3 ≈ 0.667. P(daily | kid) = 15/20 = 0.75. Inference: Kids are more likely to exercise daily (0.75 > 0.667).