Probability
Independent Events
Grade 12
Question:
<p>A problem in mathematics is given to three students \(A\), \(B\), \(C\) and their respective probability of solving the problem is 1/2, 1/3, and 1/4. Probability that the problem is solved is</p>
<p>(1) 3/4</p>
<p>(2) 1/2</p>
<p>(3) 2/3</p>
<p>(4) 1/3</p>
Step-by-Step Solution
Key Concept: Use the complement principle: find the probability that ALL students fail, then subtract from 1. This avoids complex casework of 'at least one solves'.
<p><strong>Step 1:</strong> Identify the event. 'Problem is solved' means at least one student solves it.</p><p><strong>Step 2:</strong> Use complement. P(problem solved) = 1 − P(all students fail)</p><p><strong>Step 3:</strong> Find P(all fail). Since solving is independent:</p><p>P(A fails) = 1 − 1/2 = 1/2</p><p>P(B fails) = 1 − 1/3 = 2/3</p><p>P(C fails) = 1 − 1/4 = 3/4</p><p><strong>Step 4:</strong> Multiply (independence):</p><p>P(all fail) = (1/2) × (2/3) × (3/4) = 6/24 = 1/4</p><p><strong>Step 5:</strong> Apply complement:</p><p>P(problem solved) = 1 − 1/4 = <strong>3/4</strong></p><p>∴ Answer: A</p>
Correct Answer: A