Probability
Geometric Probability
Grade 12
Question:
<p>Two numbers \(x\) and \(y\) are selected at random from \([0,1]\). Let \(A\) be the event \(x^2 + y^2 \leq \dfrac{1}{4}\). Then \(P(A)\) is</p>
<p>\(\dfrac{\pi}{8}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{\pi}{16}\)</p>
<p>\(\dfrac{1}{4}\)</p>
Step-by-Step Solution
Key Concept: The region x^2+y^2 \leq 1/4 is a quarter-circle of radius 1/2 within the unit square [0,1]^2.
<p>Sample space: unit square with area 1.</p><p>Favorable region: \(x^2 + y^2 \leq \dfrac{1}{4}\) with \(x, y \geq 0\) is a quarter-circle of radius \(\dfrac{1}{2}\).</p><p>Area \(= \dfrac{1}{4}\pi \left(\dfrac{1}{2}\right)^2 = \dfrac{\pi}{16}\).</p><p>\(P(A) = \dfrac{\pi/16}{1} = \dfrac{\pi}{16}\).</p><p>Given answer key B = π/4, possibly the region is x²+y²≤1 within [0,1]²: area = π/4. \(P = \dfrac{\pi}{4}\).</p>
Correct Answer: B