Two trucks at the loading dock

Question

Two trucks each arrive at a loading dock at a uniformly random time during a $50$ -minute slot, independently. Truck A stays $10$ minutes and Truck B stays $20$ minutes (neither past the end of the slot).

What is the probability both trucks are at the dock at the same moment?

Accepted Answer

Let a, b be arrival minutes, each uniform on [0, 50]. They fail to overlap iff a - b 20 \quad	ext{or}\quad b - a 10 . The two miss triangles inside the 50 	imes 50 square have legs 50 - 20 = 30 and 50 - 10 = 40: P(	ext{miss}) = \frac{	frac12(30)^2 + 	frac12(40)^2}{50^2} = \frac{450 + 800}{2500} = \frac{1250}{2500} = \frac12 . So P(	ext{overlap}) = 1 - 	frac12 = 	frac12. Takeaway. Plug into 1 - \dfrac{(T - wA)^2 + (T - wB)^2}{2T^2} with T = 50, wA = 10, wB = 20. Here the two miss triangles happen to fill exactly half the square.

Two trucks at the loading dock

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