The train schedule in Hummut is hopelessly unreliable. Train $A$ will enter Intersection $X$ from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train $B$ will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train $A$; again, each moment in the interval is equally likely. If each train takes 45 minutes to clear the intersection, what is the probability of a collision today?