Two airplanes are supposed to park at the same gate of a concourse. The arrival times of the airplanes are independent and randomly distributed throughout the 24 hours of the day. What is the probability that both can park at the gate, provided that the first to arrive will stay for a period of two hours, while the second can wait behind it for a period of one hour?
The set of possible events is modeled by the square $[0, 24] \times[0,24]$. It is, however, better to identify the 0th and the 24th hours, thus obtaining a square with opposite sides identified, an object that in mathematics is called a torus (which is, in fact, the Cartesian product of two circles. The favorable region is outside a band of fixed thickness along the curve $x=y$ on the torus as depicted in Figure 110. On the square model this region is obtained by removing the points $(x, y)$ with $|x-y| \leq 1$ together with those for which $|x-y-1| \leq 1$ and $|x-y+1| \leq 1$. The required probability is the ratio of the area of the favorable region to the area of the square, and is$$ P=\frac{24^{2}-2 \cdot 24}{24^{2}}=\frac{11}{12} \approx 0.917 . $$