Question

Let \( A \) and \( B \) be two events such that \( P(B) = \frac{3}{4} \) and \( P(A \cup B^C) = \frac{1}{2} \). If \( A \) and \( B \) are independent, then \( P(A) \) equals _________ (round off to 2 decimal places).

Hint:

For independent events, use the fact that \( P(A \cap B) = P(A) \times P(B) \) to simplify calculations.

Answer 1

Correct Answer: 0.32

Step 1: Use the formula for the union of events. We know that: \[ P(A \cup B^C) = P(A) + P(B^C) - P(A \cap B^C). \] Step 2: Calculate the required probabilities. Since \( P(B^C) = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4} \), and \( A \) and \( B \) are independent, we have: \[ P(A \cap B^C) = P(A) \times P(B^C) = P(A) \times \frac{1}{4}. \] Step 3: Plug the values into the formula. \[ \frac{1}{2} = P(A) + \frac{1}{4} - P(A) \times \frac{1}{4}. \] Step 4: Solve for \( P(A) \). Rearrange the equation: \[ \frac{1}{4} = P(A) \times \frac{3}{4}, \] \[ P(A) = \frac{1}{3}. \] Thus, \( P(A) = 0.33 \).