AI Tutor
You and a friend agree to meet at a café between 12:00 PM and 1:00 PM, but neither of you is willing to wait more than 15 minutes before giving up and leaving. What are the chances that you actually meet?
Because arrival times occur across a continuous spectrum of infinite possibilities, we cannot simply count outcomes—we must measure geometric area.

The 2D Geometric Probability Principle

When two continuous variables are chosen independently at uniform random, their joint outcomes form a two-dimensional sample space region (SS) on the coordinate plane.
The probability of an event EE is the ratio of the favorable area to the total sample space area: P(E)=Area(E)Area(S)P(E) = \frac{\text{Area}(E)}{\text{Area}(S)}
In 1777, French naturalist Georges-Louis Leclerc, Comte de Buffon, first proved that geometry could solve probability problems when he modeled needles dropped onto lined floorboards.
How do we translate a real-world condition like "waiting at most 15 minutes" into a shape on a coordinate grid?

Modeling Joint Arrival Times

Let xx represent your arrival time in minutes past 12:00 PM, and let yy represent your friend's arrival time, where both x,y∈[0,60]x, y \in [0, 60].
The total sample space SS is a square of dimensions 60×6060 \times 60, with an area of Area(S)=602=3600 min2\text{Area}(S) = 60^2 = 3600\text{ min}^2.
You meet if and only if the time gap between arrivals is within 15 minutes, represented by the absolute value inequality ∣x−y∣≤15|x - y| \le 15.
Removing the absolute value gives a bounded strip between two parallel lines: −15≤x−y≤15-15 \le x - y \le 15, which rearranges to x−15≤y≤x+15x - 15 \le y \le x + 15.