Probability
Independent Events
Grade 12
Question:
<p>Four persons independently solve a certain problem correctly with probabilities \(\dfrac{1}{2}, \dfrac{3}{4}, \dfrac{1}{4}, \dfrac{1}{8}\). Then the probability that the problem is solved correctly by at least one of them is</p>
<p>\(\dfrac{235}{256}\)</p>
<p>\(\dfrac{21}{256}\)</p>
<p>\(\dfrac{3}{256}\)</p>
<p>\(\dfrac{253}{256}\)</p>
Step-by-Step Solution
Key Concept: Use the complement rule: P(at least one solves) = 1 - P(none solve). This avoids summing multiple cases and is computationally cleaner.
<p><strong>Step 1:</strong> Identify the probabilities of solving correctly:</p><p>P(Person 1 solves) = 1/2, P(Person 2 solves) = 3/4, P(Person 3 solves) = 1/4, P(Person 4 solves) = 1/8</p><p><strong>Step 2:</strong> Find probabilities of NOT solving:</p><p>P(Person 1 fails) = 1 - 1/2 = 1/2</p><p>P(Person 2 fails) = 1 - 3/4 = 1/4</p><p>P(Person 3 fails) = 1 - 1/4 = 3/4</p><p>P(Person 4 fails) = 1 - 1/8 = 7/8</p><p><strong>Step 3:</strong> Since all four solve independently, probability that ALL fail:</p><p>P(all fail) = (1/2) × (1/4) × (3/4) × (7/8) = 21/256</p><p><strong>Step 4:</strong> Use complement rule:</p><p>P(at least one solves) = 1 - P(all fail) = 1 - 21/256 = <strong>235/256</strong></p><p>∴ Answer: D</p>
Correct Answer: D