Geometric Probability

Geometric probability replaces counting with measuring areas when outcomes are continuous and uniformly distributed, so probability equals favourable area divided by total area. This area-based modelling is a hallmark of harder JEE Advanced probability. Let the two arrival times x and y, in minutes after five o'clock, range over the 60 × 60 square of area 3600. The friends meet when their arrival times differ by at most 15 minutes, that is |x − y| ≤ 15. The non-meeting region consists of two right triangles where |x − y| > 15, each with legs of length 60 − 15 = 45, so each triangle has area (1/2)(45)(45) = 1012.5, totalling 2025. The favourable meeting area is 3600 − 2025 = 1575, giving probability 1575/3600 = 7/16. Option 1/4 wrongly squares the single ratio 15/60. Option 9/16 reports the complement, the non-meeting probability. Option 1/2 ignores the geometry entirely. The method uses the uniform-area definition of probability over the square sample space. Plausibility check: 7/16 ≈ 0.44 is below one-half, sensible because a 15-minute window within a 60-minute interval gives less than even odds of overlap.