2014
Problem - 2550
A girl and a guy are going to arrive at a train station. If they arrive within $10$ minutes of each other, they will instantly fall in love and live happily ever after. But after $10$ minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between $8$ AM and $9$ AM with equal probability. The guy will arrive between $7$ AM and $8:30$ AM, also with equal probability. Find the probability that the probability that they fall in love.
This problem can be solved using geometric probability technique. As shown in the diagram below. The area where they have overlap the rectangle marked with thick board. Its area is $24$ (each unit represents $10$ minutes). The area representing the case they will fall in love is shadowed with black. Its area is $6$. Additionally, we also need to consider the probability of the side length of this rectangle verse their possible arrival time. (For example, when the guy arrival before $7:50$, there is no chance he will fall in love with the girl.) Therefore, the final answer is $$\frac{6}{24}\times\frac{4}{9}\times 1=\boxed{\frac{1}{9}}$$
