Both random numbers stay small

Question

Two numbers X and Y are chosen independently and uniformly at random from [0, 1]. What is the probability that the larger of the two is at most 	frac23?

Accepted Answer

The larger of X and Y is at most 	frac23 exactly when both are at most 	frac23: \max(X, Y) \le 	frac23 \iff X \le 	frac23 	ext{ and } Y \le 	frac23 . In the unit square this is a smaller square of side 	frac23 anchored at the origin, with area P = \left(	frac23ight)^2 = \frac{4}{9} . Takeaway. A cap on the maximum splits into independent caps on each variable: P(\max \le t) = t^2 for two uniforms. It is a corner square, not a triangle.

Both random numbers stay small

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