Question

If \(A\) and \(B\) are events of a random experiment such that \[ P(A \cup B) = \frac{3}{4}, \quad P(A \cap B) = \frac{1}{4}, \quad P(\overline{A}) = \frac{2}{3}, \] then \(P(\overline{A} \cap B)\) is:

• A. \(\frac{5}{8}\)
• B. \(\frac{5}{12}\)
• C. \(\frac{3}{8}\)
• D. \(\frac{2}{5}\)

Hint:

Use properties of union and intersection to find unknown event probabilities.

Answer 1

The Correct Option is B

Step 1: Use probability formula
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Step 2: Find \(P(A)\)
\[ P(\overline{A}) = \frac{2}{3} \implies P(A) = 1 - \frac{2}{3} = \frac{1}{3} \] Step 3: Find \(P(B)\)
\[ \frac{3}{4} = \frac{1}{3} + P(B) - \frac{1}{4} \implies P(B) = \frac{3}{4} - \frac{1}{3} + \frac{1}{4} = \frac{2}{3} \] Step 4: Find \(P(\overline{A} \cap B)\)
\[ P(\overline{A} \cap B) = P(B) - P(A \cap B) = \frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12} \]