For any two events, A and B, it is given that P(A and B) = 0,12, P(A or B) = 0,83 and P(B) = 4 P(A). Are events A and B mutually exclusive? Justify your answer. Calculate P(B). Calculate P(not A).

9.2.1 Are events A and B mutually exclusive? Justify your answer. Step 1: Understand the condition for mutually exclusive events. Events A and B are mutually exclusive if the probability of both events occurring, P(A and B), is 0. Step 2: Compare the given probability with the condition. We are given P(A and B) = 0.12. Since P(A and B) = 0.12 ≠ 0, the events are not mutually exclusive. No, events A and B are not mutually exclusive. This is because P(A and B) = 0.12, which is not equal to 0. 9.2.2 Calculate P(B). Step 1: Write down the Addition Rule for probabilities. The Addition Rule states: P(A or B) = P(A) + P(B) - P(A and B) Step 2: Substitute the given values into the formula. We are given P(A or B) = 0.83 and P(A and B) = 0.12. We also know P(B) = 4P(A), which implies P(A) = (P(B))/(4). 0.83 = (P(B))/(4) + P(B) - 0.12 Step 3: Solve the equation for P(B). First, add 0.12 to both sides: 0.83 + 0.12 = (P(B))/(4) + P(B) 0.95 = (1)/(4)P(B) + 1P(B) Combine the terms with P(B): 0.95 = ((1)/(4) + (4)/(4))P(B) 0.95 = (5)/(4)P(B) Multiply both sides by (4)/(5) to find P(B): P(B) = 0.95 × (4)/(5) P(B) = 0.19 × 4 P(B) = 0.76 9.2.3 Calculate P(not A). Step 1: Find P(A) using the relationship P(B) = 4P(A). From the previous calculation, we found P(B) = 0.76. P(A) = (P(B))/(4) P(A) = (0.76)/(4) P(A) = 0.19 Step 2: Calculate P(not A) using the complement rule. The probability of an event not occurring is 1 minus the probability of the event occurring: P(not A) = 1 - P(A) P(not A) = 1 - 0.19 P(not A) = 0.81 Drop the next question! 📸