<p>Two numbers \(x\) and \(y\) are selected at random from \([0, 1]\). The probability that \(|x - y| \leq \dfrac{1}{2}\) is</p>
Step-by-Step Solution
Key Concept: The complement |x-y| > 1/2 consists of two triangles in [0,1]^2 each of area 1/8. P(complement) = 1/4.
<p>Sample space: unit square [0,1]² with area 1.</p><p>Region \(|x-y| > \dfrac{1}{2}\): two triangles with vertices at (0, 1/2),(0,1),(1/2,1) and (1,0),(1,1/2),(1/2,0).</p><p>Each triangle has area \(\dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{8}\). Total complement area \(= \dfrac{1}{4}\).</p><p>\(P(|x-y| \leq \tfrac{1}{2}) = 1 - \dfrac{1}{4} = \dfrac{3}{4}\)</p>
Correct Answer: C