A crumb near the edge of a square table

Question

A crumb falls onto a square table 10 in by 10 in, landing at a uniformly random point. What is the probability that it lands within 1 in of an edge?

Accepted Answer

Points more than 1 in from every edge form an inner square: (10 - 2)(10 - 2) = 8 	imes 8 = 64 	ext{ in}^2 . The table area is 10 	imes 10 = 100 	ext{ in}^2, so the interior fraction is 	frac{64}{100} and P(	ext{within 1 in of an edge}) = 1 - \frac{64}{100} = \frac{36}{100} = \frac{9}{25} . Takeaway. For a square of side s and strip width w, the edge probability is 1 - \left(	frac{s - 2w}{s}ight)^2. Here 1 - (0.8)^2 = 	frac{9}{25}.

A crumb near the edge of a square table

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