Question

If for two events \( A \) and \( B \), \( P(A - B) = \frac{1}{5} \) and \( P(A) = \frac{3}{5} \), then \( P(B/A) = \)

• A. \( \frac{2}{3} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{2}{5} \)

Hint:

For conditional probability, remember the formula \( P(B/A) = \frac{P(A \cap B)}{P(A)} \), and break down the events using the set operations.

Answer 1

The Correct Option is A

Step 1: Understand the given information.
We are given the following:
- \( P(A - B) = \frac{1}{5} \), which means the probability of event \( A \) occurring without event \( B \).
- \( P(A) = \frac{3}{5} \), the probability of event \( A \) occurring.

Step 2: Relationship between probabilities.
We know that:
\[ P(A) = P(A - B) + P(A \cap B) \]
This means the probability of \( A \) can be written as the sum of the probability of \( A \) without \( B \) and the probability of \( A \) and \( B \) occurring together.

Step 3: Substitute known values.
Substitute the given values into the equation:
\[ \frac{3}{5} = \frac{1}{5} + P(A \cap B) \] Solving for \( P(A \cap B) \):
\[ P(A \cap B) = \frac{3}{5} - \frac{1}{5} = \frac{2}{5} \]

Step 4: Use conditional probability formula.
We want to find \( P(B/A) \), the probability of \( B \) occurring given that \( A \) has occurred. The formula for conditional probability is:
\[ P(B/A) = \frac{P(A \cap B)}{P(A)} \]

Step 5: Substitute values into the formula.
Substitute the known values of \( P(A \cap B) \) and \( P(A) \) into the formula:
\[ P(B/A) = \frac{\frac{2}{5}}{\frac{3}{5}} = \frac{2}{3} \]

Step 6: Conclusion.
Thus, the value of \( P(B/A) \) is \( \frac{2}{3} \), which corresponds to option (A).