Question

Given the following probabilities: \[ P(A) = 0.7, P(B) = 0.5, P(A \cup B) = 0.9, \quad \text{find} \quad P(A/B) \]

• A. \( 0.2 \)
• B. \( 0.4 \)
• C. \( 0.6 \)
• D. \( 0.9 \)

Hint:

To find the conditional probability \( P(A/B) \), use the formula \( P(A/B) = \frac{P(A \cap B)}{P(B)} \). First, calculate \( P(A \cap B) \) using the formula for the union of two events.

Answer 1

The Correct Option is C

We are given the probabilities: - \( P(A) = 0.7 \), - \( P(B) = 0.5 \), - \( P(A \cup B) = 0.9 \). We need to find \( P(A/B) \), the conditional probability of \( A \) given \( B \). ### Step 1: Use the Formula for Conditional Probability The formula for conditional probability is: \[ P(A/B) = \frac{P(A \cap B)}{P(B)} \] First, use the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the known values: \[ 0.9 = 0.7 + 0.5 - P(A \cap B) \] Solve for \( P(A \cap B) \): \[ P(A \cap B) = 0.7 + 0.5 - 0.9 = 0.3 \] Now, calculate \( P(A/B) \): \[ P(A/B) = \frac{P(A \cap B)}{P(B)} = \frac{0.3}{0.5} = 0.6 \] Thus, the correct answer is: \[ \boxed{(C) 0.6} \]