Question

If P(A) = 0.3, P(B) = 0.4, then find P(A/B) if A and B are independent events.

Hint:

A key property of independent events is that the conditional probability of one event given the other is simply the probability of the first event. That is, if A and B are independent, \( P(A|B) = P(A) \) and \( P(B|A) = P(B) \). Knowing this can provide a direct answer without calculation.

Answer 1

Step 1: Understanding the Concept:
We need to find the conditional probability of event A given that event B has occurred, denoted as P(A/B). We are given that the events A and B are independent.
Step 2: Key Formula or Approach:
The definition of conditional probability is:
\[ P(A/B) = \frac{P(A \cap B)}{P(B)} \] The definition of independent events is:
\[ P(A \cap B) = P(A) \cdot P(B) \] Step 3: Detailed Explanation or Calculation:
For independent events, the occurrence of one event does not affect the probability of the other. We can show this using the formulas.
Substitute the formula for independent events into the conditional probability formula:
\[ P(A/B) = \frac{P(A) \cdot P(B)}{P(B)} \] Since \( P(B) = 0.4 \neq 0 \), we can cancel \( P(B) \) from the numerator and denominator:
\[ P(A/B) = P(A) \] We are given that \( P(A) = 0.3 \).
Therefore, \( P(A/B) = 0.3 \).
Step 4: Final Answer:
The value of P(A/B) is 0.3.