Answer: 0.1

## Explanation We are given: - P(A) = 0.50 - P(B) = 0.40 - P(A ∪ B) = 0.85 We need to find P(B | A). ### Step 1: Use the formula for union of two events The formula for the probability of the union of two events is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] ### Step 2: Solve for P(A ∩ B) Substitute the given values: \[ 0.85 = 0.50 + 0.40 - P(A \cap B) \] \[ 0.85 = 0.90 - P(A \cap B) \] \[ P(A \cap B) = 0.90 - 0.85 = 0.05 \] ### Step 3: Use the conditional probability formula The formula for conditional probability is: \[ P(B | A) = \frac{P(A \cap B)}{P(A)} \] ### Step 4: Calculate P(B | A) \[ P(B | A) = \frac{0.05}{0.50} = 0.10 \] ### Step 5: Alternative approach (as shown in the solution) We can also use the formula: \[ P(A \cup B) = P(A) + P(B) - P(B | A) \cdot P(A) \] Let x = P(B | A): \[ 0.85 = 0.50 + 0.40 - x \cdot 0.50 \] \[ 0.85 = 0.90 - 0.50x \] \[ 0.50x = 0.90 - 0.85 \] \[ 0.50x = 0.05 \] \[ x = 0.10 \] Therefore, P(B | A) = 0.10, which corresponds to option B.

Author: Nikitesh Somanthe