Question

If P(A) = 0.12, P(B) = 0.15 and P(B/A) = 0.18, then find the value of P(A \( \cap \) B).

Hint:

In conditional probability problems, identify what information is given and what is required. Notice that the value of \( P(B) \) was extra information not needed to solve for \( P(A \cap B) \) using \( P(B|A) \). Always select the correct formula based on the given conditional probability.

Answer 1

Step 1: Understanding the Concept:
This problem involves conditional probability. The notation \( P(B|A) \) represents the probability of event B occurring given that event A has already occurred. The relationship between conditional probability, joint probability, and individual probability is defined by the multiplication rule of probability.
Step 2: Key Formula or Approach:
The formula for the conditional probability of B given A is: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] To find the probability of the intersection of A and B, \( P(A \cap B) \), we can rearrange this formula: \[ P(A \cap B) = P(B|A) \times P(A) \] Step 3: Detailed Calculation:
We are given the following values:
\( P(A) = 0.12 \)
\( P(B) = 0.15 \)
\( P(B|A) = 0.18 \)
Using the rearranged formula: \[ P(A \cap B) = P(B|A) \times P(A) \] Substitute the given values into the formula: \[ P(A \cap B) = 0.18 \times 0.12 \] Now, we perform the multiplication: \[ 18 \times 12 = 216 \] Since there are a total of four decimal places in 0.18 and 0.12, the result will have four decimal places. \[ P(A \cap B) = 0.0216 \] Step 4: Final Answer:
The value of \( P(A \cap B) \) is 0.0216.