All three pieces at least a quarter long

Question

A stick of length 1 is broken at two points chosen independently and uniformly at random. What is the probability that all three pieces have length at least 	frac14?

Accepted Answer

Let the breaks be X, Y \sim U(0,1). Condition on X < Y (probability 	frac12; the case Y < X is symmetric). The three pieces are X, Y - X, and 1 - Y, and we need each at least 	frac14: X \ge 	frac14, \qquad Y - X \ge 	frac14, \qquad 1 - Y \ge 	frac14 \ \Leftrightarrow\ Y \le 	frac34. So 	frac14 \le X and X + 	frac14 \le Y \le 	frac34. The left piece must also leave room, forcing X \le 	frac12. The area of this region is \int{1/4}^{1/2} \left[	frac34 - \left(X + 	frac14ight)ight] dX = \int{1/4}^{1/2} \left(	frac12 - Xight) dX = \left[	frac{X}{2} - 	frac{X^2}{2}ight]{1/4}^…

All three pieces at least a quarter long

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