A random rectangle under a natural threshold

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 less than 	frac1e? (Here e \approx 2.718.)

Accepted Answer

Use the general result P(XY < c) = c(1 - \ln c) for 0 < c \le 1, with c = 	frac1e. Since \ln\left(	frac1eight) = -1, P\left(XY < 	frac1eight) = \frac1e\left(1 - (-1)ight) = \frac1e \cdot 2 = \frac{2}{e} \approx 0.736 . Takeaway. The threshold c = 	frac1e makes the logarithm exactly -1, collapsing c(1 - \ln c) to the clean value 	frac2e. Choosing a natural constant can turn a messy formula into a tidy answer.

A random rectangle under a natural threshold

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