How close to the edge did it land?

Question

A coin is dropped onto a rectangular rug that measures $12$ ft by $8$ ft, landing at a point uniformly at random on the rug (treat the coin as a point).

What is the probability that it lands within $1$ ft of some edge?

Accepted Answer

The points that are more than 1 ft from every edge form an inner rectangle shrunk by 1 ft on all four sides: (12 - 2)(8 - 2) = 10 	imes 6 = 60 	ext{ ft}^2 . The whole rug has area 12 	imes 8 = 96 	ext{ ft}^2. So the probability of landing in that safe interior is 	frac{60}{96}, and P(	ext{within 1 ft of an edge}) = 1 - \frac{60}{96} = \frac{36}{96} = \frac{3}{8} . Takeaway. A "within distance w of the boundary" question is the whole area minus the inner rectangle with each side reduced by 2w. Subtract the interior, do not try to add up the four strips (they overlap at the corners).

How close to the edge did it land?

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