Two random numbers with a small sum

Question

Two people each independently pick a real number uniformly at random from [0, 1]. What is the probability that their sum is at most 	frac12?

Accepted Answer

Let the picks be X and Y, plotted as a point in the unit square [0,1] 	imes [0,1], where every point is equally likely and probability equals area. The event X + Y \le 	frac12 is the region below the line X + Y = 	frac12. Within the square that line runs from (	frac12, 0) to (0, 	frac12), cutting off a right triangle with both legs 	frac12: P = 	frac12 \cdot 	frac12 \cdot 	frac12 = \frac{1}{8} . Takeaway. For two independent uniforms, any condition on X + Y becomes a region between diagonal lines in the unit square. A "sum below c" corner triangle (for c \le 1) has area 	frac{c^2}{2…

Two random numbers with a small sum

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