A random rectangle with tiny 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 less than 	frac14?

Accepted Answer

Use the general shape: P(XY < c) = c(1 - \ln c) for 0 < c \le 1. Here is the derivation for c = 	frac14. Fix X = x. If x \le 	frac14, then 	frac{1}{4x} \ge 1 and every Y works. If x 	frac14, the probability is 	frac{1}{4x}: P(XY < 	frac14) = \int0^{1/4} 1\, dx + \int{1/4}^{1} \frac{1}{4x}\, dx = \frac14 + \frac14\Big[\ln x\Big]{1/4}^{1} = \frac14 + \frac14\ln 4 . Since \ln 4 = 2\ln 2, P(XY < 	frac14) = \frac{1 + 2\ln 2}{4} \approx 0.597 . Takeaway. Plug c = 	frac14 into c(1 - \ln c). A smaller area threshold gives a smaller probability, but the log term keeps it well above the naive g…

A random rectangle with tiny area

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