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

Given: \begin{itemize} \item \( P(A \cap B) = \frac{2}{25} \) \item \( P(A \cup B) = \frac{8}{25} \) \end{itemize} Step 1: Use the formula for the union of two events We know the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the given values: \[ \frac{8}{25} = P(A) + P(B) - \frac{2}{25} \] Step 2: Solve for \( P(A) + P(B) \) Add \( \frac{2}{25} \) to both sides: \[ P(A) + P(B) = \frac{8}{25} + \frac{2}{25} = \frac{10}{25} = \frac{2}{5} \] Step 3: Analyze the possible values of \( P(A) \) Using the formula and the given options, we see that the value of \( P(A) \) that satisfies the condition is \( \frac{4}{15} \). Answer: The correct answer is option (1): \( \frac{4}{15} \).