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.
ScanSolve AI Answer
You're on a roll — Here are the solutions for Question 1: Given: P(A) = 0.8 P(A B) = 0.7 a) Find P(B') if A and B are mutually exclusive. Step 1: Understand the condition for mutually exclusive events. If events A and B are mutually exclusive, then P(A B) = 0. Step 2: Compare with the given information. The problem states that P(A B) = 0.7. Step 3: Conclude. Since P(A B) = 0.7 ≠ 0, events A and B cannot be mutually exclusive. Therefore, it is impossible for A and B to be mutually exclusive under the given conditions. b) Find P(B') if A and B are independent. Step 1: Use the condition for independent events. If events A and B are independent, then P(A B) = P(A) · P(B). Step 2: Substitute the given values to find P(B). 0.7 = 0.8 · P(B) P(B) = (0.7)/(0.8) = (7)/(8) P(B) = 0.875 Step 3: Calculate P(B'). P(B') = 1 - P(B) P(B') = 1 - 0.875 0.125 c) Find P(B') if P(A B) = 0.95. Step 1: Use the formula for the probability of the union of two events. P(A B) = P(A) + P(B) - P(A B) Step 2: Substitute the given values to find P(B). 0.95 = 0.8 + P(B) - 0.7 0.95 = 0.1 + P(B) P(B) = 0.95 - 0.1 P(B) = 0.85 Step 3: Calculate P(B'). P(B') = 1 - P(B) P(B') = 1 - 0.85 0.15