Probability
Independent Events
Grade 12
Question:
<p>The probability that a teacher will give a surprise test during any class meeting is \(\frac{1}{5}\). If a student is absent twice, then the probability that the student will miss at least one test is</p>
<p>(1) \(\dfrac{4}{5}\)</p>
<p>(2) \(\dfrac{2}{5}\)</p>
<p>(3) \(\dfrac{7}{25}\)</p>
<p>(4) \(\dfrac{9}{25}\)</p>
Step-by-Step Solution
Key Concept: Model this as independent events where the student misses a test only if a test occurs during their absence. Use the complement: P(miss at least one) = 1 - P(miss none) = 1 - P(no test on either day).
<p><strong>Step 1:</strong> Identify what we need. The student misses a test if and only if a test is given on a day when they're absent.</p><p><strong>Step 2:</strong> P(test during any class) = 1/5, so P(no test during any class) = 4/5</p><p><strong>Step 3:</strong> The student is absent on 2 separate occasions. For the student to miss NO tests, there must be no test on either of these 2 days.</p><p><strong>Step 4:</strong> P(no test on day 1 AND no test on day 2) = (4/5) × (4/5) = 16/25 (by independence)</p><p><strong>Step 5:</strong> P(miss at least one test) = 1 - P(miss no tests) = 1 - 16/25 = <strong>9/25</strong></p><p>∴ Answer: D</p>
Correct Answer: D