Probability
Classical Probability
Grade 12
Question:
<p><strong>For Problems 7–9</strong><br>A cube having all of its sides painted is cut by two horizontal, two vertical, and other two planes so as to form 27 cubes all having the same dimensions. Of these cubes, a cube is selected at random.</p><p><strong>Problem 7:</strong> The probability that the cube selected has none of its sides painted is</p>
<p>1/9</p>
<p>1/27</p>
<p>1/18</p>
<p>5/54</p>
Step-by-Step Solution
Key Concept: When a painted cube is cut into 27 equal smaller cubes (3×3×3 grid), only the innermost cube touches no original faces. Identifying which of the 27 cubes are interior (unpainted) is key.
<p><strong>Step 1:</strong> Understand the cutting pattern. A painted cube is cut by 2 horizontal + 2 vertical + 2 other planes (total 6 planes), creating a 3×3×3 arrangement of 27 identical small cubes.</p><p><strong>Step 2:</strong> Classify the 27 cubes by painted faces:</p><ul><li><strong>Corner cubes:</strong> 8 cubes with 3 painted faces</li><li><strong>Edge cubes:</strong> 12 cubes with 2 painted faces</li><li><strong>Face cubes:</strong> 6 cubes with 1 painted face</li><li><strong>Center cube:</strong> 1 cube with 0 painted faces (completely interior)</li></ul><p><strong>Step 3:</strong> Verify: 8 + 12 + 6 + 1 = 27 ✓</p><p><strong>Step 4:</strong> Calculate probability. Number of cubes with no painted sides = 1. Total cubes = 27.</p><p><strong>Step 5:</strong> P(no painted sides) = 1/27</p><p>∴ Answer: B</p>
Correct Answer: B