This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to Question 1: Question 1.1 Given P(A) = 0.48 and P(B) = 0.26. Step 1: Calculate P(A or B) for mutually exclusive events. For mutually exclusive events, the probability of A or B occurring is the sum of their individual probabilities. P(A or B) = P(A) + P(B) P(A or B) = 0.48 + 0.26 P(A or B) = 0.74 Step 2: Calculate P(A or B) for independent events. For independent events, the probability of A and B occurring is the product of their individual probabilities: P(A and B) = P(A) × P(B). The general formula for P(A or B) is: P(A or B) = P(A) + P(B) - P(A and B) Substitute the formula for independent events: P(A or B) = P(A) + P(B) - (P(A) × P(B)) P(A or B) = 0.48 + 0.26 - (0.48 × 0.26) P(A or B) = 0.74 - 0.1248 P(A or B) = 0.6152 Question 1.2 A survey was conducted amongst 150 grade 11 learners. Step 1: Write down the value of x. From the table, the total number of boys is 80. The sum of boys' preferences for Netball, Soccer, and Rugby must equal this total. 32 + x + 27 = 80 59 + x = 80 x = 80 - 59 x = 21 Step 2: Calculate the probability that a learner is a boy and does not like netball. Total learners = 150. Number of boys = 80. Number of boys who like Netball = 32. Number of boys who do not like Netball = Total boys - Boys who like Netball = 80 - 32 = 48 The probability is the number of boys who do not like netball divided by the total number of learners. P(boy and not netball) = (48)/(150) Simplify the fraction: P(boy and not netball) = (24)/(75) = (8)/(25) P(boy and not netball) = 0.32 Step 3: Determine if the events "being a boy" and "liking soccer" are independent. Two events A and B are independent if P(A and B) = P(A) × P(B). Let A be the event "being a boy". Let B be the event "liking soccer". Calculate the individual probabilities: Total learners = 150. Number of boys = 80. P(boy) = (80)/(150) Number of learners who like Soccer = 33. P(soccer) = (33)/(150) Number of boys who like Soccer (value of x) = 21. P(boy and soccer) = (21)/(150) Check for independence: Is P(boy and soccer) = P(boy) × P(soccer)? (21)/(150) = (80)/(150) × (33)/(150) Calculate the right side: (80)/(150) × (33)/(150) = (8)/(15) × (33)/(150) = (264)/(2250) Simplify (264)/(2250) by dividing by 6: (264 ÷ 6)/(2250 ÷ 6) = (44)/(375) Now compare this to the left side: (21)/(150) Simplify (21)/(150) by dividing by 3: (21 ÷ 3)/(150 ÷ 3) = (7)/(50) Since (7)/(50) ≠ (44)/(375) (or 0.14 ≠ 0.1173), the events are not independent. The events "being a boy" and "liking soccer" are NOT independent. That's 2 down. 3 left today — send the next one.