Probability
Classical Probability
Grade None
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 8:</strong> The probability that the cube selected has two sides painted is</p>
<p>1/9</p>
<p>4/9</p>
<p>8/27</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: When a painted cube is cut into 27 equal smaller cubes (3×3×3 division), cubes with exactly 2 painted faces are located only on the edges (not at corners or center). Count edge cubes excluding corner cubes: there are 12 edges, each with 1 such cube = 12 cubes.
<p><strong>Step 1:</strong> Understand the structure. A cube with all sides painted is cut into 3×3×3 = 27 smaller cubes of equal dimension.</p><p><strong>Step 2:</strong> Categorize the 27 cubes by number of painted faces:</p><ul><li><strong>Corner cubes:</strong> 8 cubes with 3 painted faces (at vertices)</li><li><strong>Edge cubes:</strong> 12 cubes with 2 painted faces (middle of each edge)</li><li><strong>Face cubes:</strong> 6 cubes with 1 painted face (center of each face)</li><li><strong>Center cube:</strong> 1 cube with 0 painted faces (interior)</li></ul><p><strong>Step 3:</strong> Identify cubes with exactly 2 painted faces. These are the middle cubes on each of the 12 edges of the original cube. Count = 12.</p><p><strong>Step 4:</strong> Calculate probability:</p><p>P(exactly 2 painted faces) = Number of cubes with 2 painted faces / Total cubes = 12/27 = 4/9</p><p>∴ Answer: C (4/9)</p>
Correct Answer: C