Question

Let \( A, B \) be two events and \( \overline{A} \) be the complement of \( A \). If \( P(\overline{A}) = 0.7 \), \( P(B) = 0.7 \), and \( P(B|A) = 0.5 \), then \( P(A \cup B) = \) ...........

• A. 0.65
• B. 0.85
• C. 0.75
• D. 0.50

Hint:

To calculate the union of two events, always remember the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. If you are given conditional probabilities, use $P(B|A) = \frac{P(A \cap B)}{P(A)}$ to find $P(A \cap B)$.

Answer 1

The Correct Option is B

We know the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] We are given:
- $P(\overline{A}) = 0.7$, so $P(A) = 1 - P(\overline{A}) = 1 - 0.7 = 0.3$
- $P(B) = 0.7$
- $P(B|A) = 0.5$
Now, to calculate $P(A \cap B)$, we use the definition of conditional probability: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] Substituting the values: \[ 0.5 = \frac{P(A \cap B)}{0.3} \] Thus, \[ P(A \cap B) = 0.5 \times 0.3 = 0.15 \] Now, substitute the values into the union formula: \[ P(A \cup B) = 0.3 + 0.7 - 0.15 = 0.85 \] Thus, $P(A \cup B) = 0.85$.