Two random numbers with a large sum

Question

Two numbers X and Y are chosen independently and uniformly at random from [0, 1]. What is the probability that X + Y \le 	frac54?

Accepted Answer

Work in the unit square, where probability equals area. It is easier to find the complementary region X + Y 	frac54. The line X + Y = 	frac54 meets the square at (1, 	frac14) and (	frac14, 1), cutting off a corner triangle at (1,1) with both legs 1 - 	frac14 = 	frac34: P(X + Y 	frac54) = 	frac12 \left(	frac34ight)^2 = \frac{9}{32} . Therefore P(X + Y \le 	frac54) = 1 - \frac{9}{32} = \frac{23}{32} \approx 0.719 . Takeaway. When the line cuts near the far corner (c 1), the leftover corner triangle has legs 2 - c and area 	frac{(2-c)^2}{2}. Subtract it from one.

Two random numbers with a large sum

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