Question

If \( P(A) = \frac{3}{5} \), \( P(\overline{B}) = \frac{4}{7} \), and \( P(A \cup B) = \frac{2}{3} \), which of the following are correct?  

(A) \( P(A \cap B) = \frac{17}{105} \) 
(B) \( P(A \mid B) = \frac{17}{45} \) 
(C) A and B are independent events 
(D) \( P(B \mid A) = \frac{17}{36} \)

• A. (A), (B) and (D) only
• B. (B) and (C) only
• C. (A) and (D) only
• D. (A) and (B) only

Hint:

The definition of independent events is \( P(A \cap B) = P(A)P(B) \). Always verify this before calculating conditional probabilities.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Given: \( P(A) = \frac{3}{5} = 0.6 \).
\( P(B) = 1 - P(\bar{B}) = 1 - \frac{4}{7} = \frac{3}{7} \approx 0.428 \).
\( P(A \cup B) = \frac{2}{3} \approx 0.667 \).

Step 2: Detailed Explanation:
1. Finding \( P(A \cap B) \):
\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
\( \frac{2}{3} = \frac{3}{5} + \frac{3}{7} - P(A \cap B) \)
\( P(A \cap B) = \frac{3}{5} + \frac{3}{7} - \frac{2}{3} = \frac{63 + 45 - 70}{105} = \frac{38}{105} \).
Note: The provided statement (A) says 17/105, which contradicts the result. However, re-evaluating the provided options vs calculation.
If calculation yields \( \frac{38}{105} \), and we look at conditional probability:
\( P(A/B) = \frac{P(A \cap B)}{P(B)} = \frac{38/105}{3/7} = \frac{38}{105} \times \frac{7}{3} = \frac{38}{45} \).

Step 3: Final Answer:
Given the potential for a typo in the question's values or options, based on standard probability formulas, none of the specific sets exactly match if calculated with these values. I recommend verifying the input values.