Question

Let A and B are independent events and P(A)=0.3 and P(B)=0.4 ; then find P(B/A).

Hint:

This is a conceptual question. Recognizing that \( P(B|A) = P(B) \) for independent events provides the answer instantly without any calculation. This is a very common question type in probability.

Answer 1

Step 1: Understanding the Concept:
This question deals with conditional probability for independent events. Two events are independent if the occurrence of one does not affect the probability of the other. The notation \( P(B|A) \) represents the conditional probability of event B occurring given that event A has already occurred.
Step 2: Key Formula or Approach:
The definition of independent events A and B is that \( P(A \cap B) = P(A) \cdot P(B) \).
The formula for conditional probability is \( P(B|A) = \frac{P(A \cap B)}{P(A)} \).
For independent events, this simplifies directly to \( P(B|A) = P(B) \).
Step 3: Detailed Explanation:
Given that events A and B are independent. By the definition of independence, the probability of B occurring is not affected by whether A has occurred or not. Therefore, the probability of B given A is simply the probability of B. \[ P(B|A) = P(B) \] We are given that \( P(B) = 0.4 \). So, \[ P(B|A) = 0.4 \] Alternative Method (using formula): First, calculate \( P(A \cap B) \) using the independence rule: \[ P(A \cap B) = P(A) \cdot P(B) = (0.3)(0.4) = 0.12 \] Now, use the conditional probability formula: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.12}{0.3} = \frac{12}{30} = \frac{4}{10} = 0.4 \] Both methods yield the same result.
Step 4: Final Answer:
The value of \( P(B|A) \) is 0.4.