Probability
Independent Events
Grade 12
Question:
<p>If four persons independently solve a certain problem correctly with probabilities \(\frac{1}{2}\), \(\frac{3}{4}\), \(\frac{1}{4}\) and \(\frac{1}{8}\), then the probability that the problem is solved correctly by at least one of them is</p>
<p>(a) \(\frac{235}{256}\)</p>
<p>(b) \(\frac{21}{256}\)</p>
<p>(c) \(\frac{3}{256}\)</p>
<p>(d) \(\frac{253}{256}\)</p>
Step-by-Step Solution
Key Concept: For 'at least one' problems with independent events, use the complement: find the probability that none of the events occur.
<p><strong>Step 1:</strong> Use the complement: $P(\text{at least one solves}) = 1 - P(\text{none solve})$.</p><p><strong>Step 2:</strong> Probabilities of failing: $1 - \frac{1}{2} = \frac{1}{2}$, $1 - \frac{3}{4} = \frac{1}{4}$, $1 - \frac{1}{4} = \frac{3}{4}$, $1 - \frac{1}{8} = \frac{7}{8}$.</p><p><strong>Step 3:</strong> $P(\text{none solve}) = \frac{1}{2} \times \frac{1}{4} \times \frac{3}{4} \times \frac{7}{8} = \frac{21}{256}$.</p><p><strong>Step 4:</strong> $P(\text{at least one solves}) = 1 - \frac{21}{256} = \frac{235}{256}$.</p>
Correct Answer: A