Question

Let A and B be two events such that $ P(A/B) = \frac{1}{2}, \quad P(B/A) = \frac{1}{3}, \quad \text{and} \quad P(A \cap B) = \frac{1}{6} $ Then, which one of the following is not true?

• A. A and B are not independent
• B. \( P(A \cup B) = \frac{2}{3} \)
• C. \( P(A' \cap B) = \frac{1}{6} \)
• D. A and B are independent

Hint:

For independent events, the product of the individual probabilities should equal the joint probability: \( P(A \cap B) = P(A) \cdot P(B) \).

Answer 1

The Correct Option is A

We are given the following information: \[ P(A/B) = \frac{1}{2}, \quad P(B/A) = \frac{1}{3}, \quad P(A \cap B) = \frac{1}{6} \]
Step 1: Find \( P(A) \) and \( P(B) \)
We use the definition of conditional probability to express \( P(A/B) \) and \( P(B/A) \): \[ P(A/B) = \frac{P(A \cap B)}{P(B)} \quad \Rightarrow \quad \frac{1}{2} = \frac{\frac{1}{6}}{P(B)} \quad \Rightarrow \quad P(B) = \frac{1}{3} \] Now, using the other conditional probability: \[ P(B/A) = \frac{P(A \cap B)}{P(A)} \quad \Rightarrow \quad \frac{1}{3} = \frac{\frac{1}{6}}{P(A)} \quad \Rightarrow \quad P(A) = \frac{1}{2} \]
Step 2: Check the Independence Condition
For events A and B to be independent, the following condition must hold: \[ P(A \cap B) = P(A) \cdot P(B) \] Substituting the values of \( P(A) \) and \( P(B) \): \[ P(A \cap B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \] Since this is true, we can conclude that A and B are independent.
Step 3: Calculate \( P(A \cup B) \)
The formula for the union of two events is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the known values: \[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{2}{3} \] So, option (B) is correct.
Step 4: Find \( P(A' \cap B) \)
The probability \( P(A' \cap B) \) is given by: \[ P(A' \cap B) = P(B) - P(A \cap B) = \frac{1}{3} - \frac{1}{6} = \frac{1}{6} \] So, option (C) is correct.
Conclusion Since we have established that A and B are independent, the correct answer is:
\[ \text{A and B are not independent} \] Thus, the correct answer is (A).