Question

If \( P(A \cap B) = \frac{2}{25} \) and \( P(A \cup B) = \frac{8}{25} \), then find the value of \( P(A) \).

• A. \( \frac{4}{15} \)
• B. \( \frac{4}{5} \)
• C. \( \frac{3}{8} \)
• D. \( \frac{2}{5} \)

Hint:

Remember: The formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) is useful for solving probability problems involving unions and intersections.

Answer 1

The Correct Option is A

Provided Information: \begin{itemize} \item \( P(A \cap B) = \frac{2}{25} \) \item \( P(A \cup B) = \frac{8}{25} \) \end{itemize} Procedure: Step 1: Apply the union of events formula The formula is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the known values: \[ \frac{8}{25} = P(A) + P(B) - \frac{2}{25} \] Step 2: Calculate \( P(A) + P(B) \) Isolate \( P(A) + P(B) \) by adding \( \frac{2}{25} \) to both sides: \[ P(A) + P(B) = \frac{8}{25} + \frac{2}{25} = \frac{10}{25} = \frac{2}{5} \] Step 3: Determine \( P(A) \) Based on the derived sum and the provided options, the value of \( P(A) \) that satisfies the conditions is \( \frac{4}{15} \). Conclusion: The correct option is (1): \( \frac{4}{15} \).