GATE CS 1998 Q3a — Probability: Random Variables and Distributions

Let the arrival times of the two friends at the park be $x$ and $y$ minutes after 4:00 pm, respectively. Since they can arrive at any time between 4:00 pm and 5:00 pm, we have:

This forms a square sample space of size $60 \times 60$ with a total area of:

Each friend waits for $10$ minutes or until the end of the hour (whichever is earlier). This means they will meet if and only if the difference between their arrival times is at most $10$ minutes:

Conversely, they will not meet if:

This non-meeting region consists of two symmetric right-angled triangles in the corners of the square:

1. The lower-right triangle defined by $x - y > 10$, with vertices at $(10, 0)$, $(60, 0)$, and $(60, 50)$.
2. The upper-left triangle defined by $y - x > 10$, with vertices at $(0, 10)$, $(0, 60)$, and $(50, 60)$.

Each of these triangles has perpendicular sides of length $50$ (since $60 - 10 = 50$).

The combined area of these two triangles is:

Therefore, the probability that the two friends will not meet is:

Thus, the probability is approximately 0.69 (or 0.694).