Probability
Independent Events
Grade 12
Question:
<p>Let \(P(A)\), \(P(B)\) and \(P(C)\) denote the probability of solving a problem by \(A\), \(B\) and \(C\) respectively, where \(P(A) = \frac{1}{2}\), \(P(B) = \frac{1}{3}\) and \(P(C) = \frac{1}{4}\). The probability that the problem is solved is:</p>
<p>\(\frac{1}{4}\)</p>
<p>\(\frac{3}{4}\)</p>
<p>\(\frac{1}{2}\)</p>
<p>\(\frac{2}{3}\)</p>
Step-by-Step Solution
Key Concept: The problem is solved if at least one person solves it. Use the complement: P(solved) = 1 - P(none solve it) = 1 - P(A' ∩ B' ∩ C'). Since events are independent, multiply individual probabilities of NOT solving.
<p><strong>Step 1:</strong> Identify that the problem is solved if <strong>at least one</strong> person solves it.</p><p><strong>Step 2:</strong> Use complement: P(problem solved) = 1 - P(no one solves it)</p><p><strong>Step 3:</strong> Find probabilities of NOT solving:</p><p>P(A') = 1 - 1/2 = 1/2</p><p>P(B') = 1 - 1/3 = 2/3</p><p>P(C') = 1 - 1/4 = 3/4</p><p><strong>Step 4:</strong> Since events are independent:</p><p>P(A' ∩ B' ∩ C') = P(A') × P(B') × P(C') = (1/2) × (2/3) × (3/4) = 6/24 = 1/4</p><p><strong>Step 5:</strong> Therefore:</p><p>P(problem solved) = 1 - 1/4 = <strong>3/4</strong></p><p>∴ Answer: B</p>
Correct Answer: B