Question

If \( P(A) = \frac{6}{11} \), \( P(B) = \frac{5}{11} \), and \( P(A \cup B) = \frac{7}{11} \), then \( P(A \cap B) \) is:

• A. \( \frac{4}{11} \)
• B. \( \frac{5}{11} \)
• C. \( \frac{7}{11} \)
• D. \( \frac{9}{11} \)

Hint:

Remember the addition rule for probabilities: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] You can rearrange this to find the intersection if the union and individual probabilities are known.

Answer 1

The Correct Option is A

Step 1: Recall the formula relating union and intersection of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Step 2: Rearranging for \( P(A \cap B) \): \[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \] Step 3: Substitute the given values: \[ P(A \cap B) = \frac{6}{11} + \frac{5}{11} - \frac{7}{11} = \frac{11}{11} - \frac{7}{11} = \frac{4}{11} \] Final Answer: \( \boxed{\frac{4}{11}} \)