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
Here's the solution for question 2(i): (i) Given P(A)=(1)/(2), P(B^c)=(5)/(6), and P(A B)=(5)/(12). Step 1: Calculate P(B). P(B) = 1 - P(B^c) P(B) = 1 - (5)/(6) P(B) = (6)/(6) - (5)/(6) P(B) = (1)/(6) a) Find P(A B). Step 2: Use the Addition Rule for Probability. P(A B) = P(A) + P(B) - P(A B) Substitute the known values: (5)/(12) = (1)/(2) + (1)/(6) - P(A B) Find a common denominator for the fractions on the right side: (5)/(12) = (3)/(6) + (1)/(6) - P(A B) (5)/(12) = (4)/(6) - P(A B) Simplify (4)/(6) to (2)/(3): (5)/(12) = (2)/(3) - P(A B) Rearrange to solve for P(A B): P(A B) = (2)/(3) - (5)/(12) Find a common denominator for (2)/(3) and (5)/(12): P(A B) = (8)/(12) - (5)/(12) P(A B) = (3)/(12) P(A B) = (1)/(4) b) Find P(A B^c). Step 3: Understand the notation P(A B^c). The event A B^c means that event A occurs and event B^c does not occur. If B^c does not occur, it means event B occurs. Therefore, A B^c is equivalent to A B. P(A B^c) = P(A B) Step 4: Use the result from part (a). P(A B^c) = (1)/(4) 3 done, 2 left today. You're making progress.