Vishal and Anirudh decide to meet at a cafe between 5 pm and 6 pm. They agree that the person who arrives first at the cafe would wait for exactly 15 minutes for the other. If each of them arrives at a random time between 5 pm and 6 pm , what is the probability that the meeting takes place? (A) 9/16 (B) 1/2 (C) 7/16 (D) 5/8

Let Vishal's time be considered on the X-axis and Anirudh's time be considered on the Y-axis. The total time of consideration is a square of side 60.

If Vishal arrives at 0 minutes, Anirudh can arrive at any time before 15 min. If Vishal arrives at 1 min, Anirudh can arrive at any time before 16 min. If Vishal arrives at 15 min, then Anirudh can arrive at any time between 0 and 30 minutes. Similarly, if Vishal arrives at 16 min, Anirudh can arrive at any time between 1 and 31 min. If we draw the area of intersection carefully, we see that the required area is a hexagon with two adjacent sides of lengths 15 on X and Y axes respectively; 2 adjacent sides of 15 on the other two sides of the square and two sides of $$45\sqrt2$$ parallel to each other.

The required probability is the area of this hexagon divided by the area of the square = 1575/3600 = 63/144 = 7/16

1-9/16=7/16