In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q. The probability that any point picked randomly within the square falls in the shaded area is ____________.
A. 4 - pi
B. 1/2
C. pi/2 - 1
D. 1/4
GATE 2021 · General Aptitude · Circle · medium
Answer: Probability = pi/2 - 1 ≈ 0.571
Set up coordinates: Square has vertices at the corners of the unit square. Total area = 1.
Area of each quarter-circle sector: Each sector (at S and at Q) has area pi/4
Union of both sectors covers the entire square: Every point in the square is within distance 1 from either S or Q (or both), so the union of the two sectors exactly covers the square
Apply inclusion-exclusion to find the intersection (shaded area): Shaded area = pi/4 + pi/4 - 1 = pi/2 - 1 ≈ 0.571