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.