Question

Two independent events \( A \) and \( B \) have individual probabilities of occurring given by \( P(A) = 0.4 \) and \( P(B) = 0.5 \). Find the probability that at least one of these two events occurs.

• A. \( 0.70 \)
• B. \( 0.90 \)
• C. \( 0.20 \)
• D. \( 0.30 \)

Hint:

Never simply add independent probabilities together (\( 0.4 + 0.5 = 0.9 \)) without checking for overlap. Unless the problem explicitly states that the events are mutually exclusive, you must always calculate and subtract their intersection point.

Answer 1

The Correct Option is A

Concept: The phrase "probability of occurrence of at least one event" refers to the union of the two sets, represented as \( P(A \cup B) \). According to the probability addition theorem: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] For independent events, their joint intersection is the direct product of their separate individual probabilities: \[ P(A \cap B) = P(A) \cdot P(B) \]

Step 1: Calculate the intersection probability for the independent events.
Since events \( A \) and \( B \) are explicitly stated to operate independently, calculate their overlapping intersection: \[ P(A \cap B) = P(A) \cdot P(B) = 0.4 \times 0.5 = 0.20 \]

Step 2: Apply values to the probability addition theorem.
Plug the individual parameters along with our calculated intersection value into the addition formula: \[ P(A \cup B) = 0.4 + 0.5 - 0.20 \] Simplify the addition and subtraction steps: \[ P(A \cup B) = 0.90 - 0.20 = 0.70 \]