Bouncing near the padded edge

Question

A jumper lands at a uniformly random point on a square trampoline that is $8$ ft by $8$ ft. A padded border runs $2$ ft inward from every edge.

What is the probability that the jumper lands on the padded border (within $2$ ft of an edge)?

Accepted Answer

The un-padded center, more than 2 ft from every edge, is an inner square: (8 - 4)(8 - 4) = 4 	imes 4 = 16 	ext{ ft}^2 . The trampoline area is 8 	imes 8 = 64 	ext{ ft}^2, so the center fraction is 	frac{16}{64} = 	frac14 and P(	ext{on the padded border}) = 1 - \frac14 = \frac34 . Takeaway. When the strip width reaches a quarter of the side, the safe center is only \left(	frac12ight)^2 = 	frac14 of the area, so three quarters of landings are near the edge.

Bouncing near the padded edge

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