<p>Two students A and B solve a problem. Their respective probabilities of solving the problem are \(\frac{2}{3}\) and \(\frac{1}{2}\). The probability of the problem being solved by at least one of them is ______.</p>
Step-by-Step Solution
Key Concept: Use the complement rule: P(at least one solves) = 1 - P(both fail). This avoids complicated casework and is computationally cleaner than adding individual probabilities.
<p><strong>Step 1:</strong> Identify the complementary event. The complement of 'at least one solves' is 'both fail'.</p><p><strong>Step 2:</strong> Find P(A fails) and P(B fails). Since P(A solves) = 2/3, we have P(A fails) = 1/3. Since P(B solves) = 1/2, we have P(B fails) = 1/2.</p><p><strong>Step 3:</strong> Assuming independence, P(both fail) = P(A fails) × P(B fails) = (1/3) × (1/2) = 1/6.</p><p><strong>Step 4:</strong> Apply the complement rule: P(at least one solves) = 1 - P(both fail) = 1 - 1/6 = 5/6.</p><p>∴ Answer: <strong>5/6</strong></p>
Correct Answer: 5