<p><strong>For Problems 1–3:</strong> In a class of 10 students, probability of exactly <i>i</i> students passing an examination is directly proportional to <i>i</i><sup>2</sup>. Then answer the following questions:</p><p>The probability that exactly 5 students passing an examination is</p>
Step-by-Step Solution
Key Concept: Since P(i) ∝ i², we have P(i) = ki² for some constant k. Use the fact that all probabilities sum to 1 to find k, then calculate P(5) = k(5²).
<p><strong>Step 1:</strong> Since P(exactly i students pass) ∝ i², we write P(i) = ki² where k is a constant.</p><p><strong>Step 2:</strong> The sum of all probabilities equals 1:</p><p>$$\sum_{i=0}^{10} P(i) = 1$$</p><p>$$k\sum_{i=0}^{10} i^2 = 1$$</p><p><strong>Step 3:</strong> Calculate $\sum_{i=0}^{10} i^2 = 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385$</p><p><strong>Step 4:</strong> Therefore: $k \cdot 385 = 1$, so $k = \frac{1}{385}$</p><p><strong>Step 5:</strong> The probability that exactly 5 students pass:</p><p>$$P(5) = k \cdot 5^2 = \frac{1}{385} \cdot 25 = \frac{25}{385} = \frac{5}{77}$$</p><p>∴ Answer: $\frac{5}{77}$ (Option C)</p>
Correct Answer: C