A random rectangle with small 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 	frac12?

Accepted Answer

Work in the unit square where probability equals area, and find P(XY < 	frac12). Fix X = x. Then XY < 	frac12 needs Y < 	frac{1}{2x}. If x \le 	frac12, the bound 	frac{1}{2x} \ge 1 so every Y works (probability 1). If x 	frac12, the probability is 	frac{1}{2x}. Averaging over x: P(XY < 	frac12) = \int0^{1/2} 1 \, dx + \int{1/2}^{1} \frac{1}{2x}\, dx = \frac12 + \frac12\Big[\ln x\Big]{1/2}^{1} = \frac12 + \frac12\ln 2 . So P(XY < 	frac12) = \frac{1 + \ln 2}{2} \approx 0.847 . Takeaway. The region XY < c sits under a hyperbola in the unit square. For 0 < c \le 1, its area is c(1 - \ln …

A random rectangle with small area

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