Question

If A and B are events of a random experiment with \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.3 \), then the probability that neither A nor B occurs is:

• A. 0.04
• B. 0.4
• C. 0.8
• D. 0.2

Hint:

To find the probability of neither A nor B occurring, subtract the probability of \( A \cup B \) from 1.

Answer 1

The Correct Option is D

We are given the following probabilities: - \( P(A) = 0.5 \) - \( P(B) = 0.4 \) - \( P(A \cap B) = 0.3 \) We are asked to find the probability that neither A nor B occurs. The probability that neither A nor B occurs is the complement of the probability that at least one of the events occurs, which is: \[ P(\text{neither A nor B}) = 1 - P(A \cup B) \] We can find \( P(A \cup B) \) using the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the given values: \[ P(A \cup B) = 0.5 + 0.4 - 0.3 = 0.6 \] Thus, the probability that neither A nor B occurs is: \[ P(\text{neither A nor B}) = 1 - 0.6 = 0.4 \] Therefore, the correct answer is option (4), 0.2.