Two students A and B have independent failure probabilities in an exam: P(A fails) = 0.2 and P(B fails) = 0.3. What is the probability that at least one of A or B fails (i.e., A fails or B fails or both)?

Introduction / Context:This is a union-of-events problem with independence. We want P(A ∪ B) where A = “A fails” and B = “B fails.”

Given Data / Assumptions:

• P(A) = 0.2, P(B) = 0.3.
• Independence: P(A ∩ B) = P(A) * P(B) = 0.06.

Concept / Approach:Use inclusion–exclusion for two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Step-by-Step Solution:Compute the intersection: 0.2 × 0.3 = 0.06.Therefore P(A ∪ B) = 0.2 + 0.3 − 0.06 = 0.44.

Verification / Alternative check:Complement method: P(neither fails) = (0.8)(0.7) = 0.56; hence P(at least one fails) = 1 − 0.56 = 0.44 (same result).

Why Other Options Are Wrong:0.38 is not obtained by any standard combination; 0.50 ignores overlap subtraction; 0.94 is near certainty and clearly too large; 0.56 is the complement (both pass), not the asked probability.

Common Pitfalls:Adding probabilities without subtracting the overlap when events are not mutually exclusive.

Final Answer:0.44