A random rectangle with large area

Question

Two side lengths X and Y are each chosen independently and uniformly at random from [0, 1], forming a rectangle with area XY. What is the probability the rectangle's area is greater than 	frac12?

Accepted Answer

Use the complement. For two uniforms, P(XY < c) = c(1 - \ln c), so at c = 	frac12: P(XY < 	frac12) = 	frac12\left(1 - \ln 	frac12ight) = 	frac12(1 + \ln 2) = \frac{1 + \ln 2}{2} . Therefore P(XY 	frac12) = 1 - \frac{1 + \ln 2}{2} = \frac{1 - \ln 2}{2} \approx 0.153 . Takeaway. A big product is hard for two numbers under 1: both must be large at once. Only about 15\% of random rectangles clear half the unit area.

A random rectangle with large area

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