Both random numbers stay large

Question

Two numbers X and Y are chosen independently and uniformly at random from [0, 1]. What is the probability that the smaller of the two is at least 	frac14?

Accepted Answer

The smaller of X and Y is at least 	frac14 exactly when both are at least 	frac14: \min(X, Y) \ge 	frac14 \iff X \ge 	frac14 	ext{ and } Y \ge 	frac14 . In the unit square this is the top-right square with each side running from 	frac14 to 1, side length 	frac34: P = \left(	frac34ight)^2 = \frac{9}{16} . Takeaway. A floor on the minimum means both variables clear the floor: P(\min \ge t) = (1 - t)^2 for two uniforms. Here (1 - 	frac14)^2 = 	frac{9}{16}.

Both random numbers stay large

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