One cut, how lopsided are the two pieces?

Question

A ribbon of length 1 is cut at a single point chosen uniformly at random along its length, producing two pieces. What is the probability that the longer piece is at least 3 times as long as the shorter piece?

Accepted Answer

Let the cut fall at position U, uniform on [0, 1]. The two pieces have lengths U and 1 - U. By symmetry, first suppose U \le 	frac12, so the shorter piece is U and the longer is 1 - U. "Longer is at least 3 times shorter" means 1 - U \ge 3U \quad\Longrightarrow\quad U \le 	frac14 . So on the left half the cut must land in [0, 	frac14], a length of 	frac14. The mirror case U 	frac12 contributes the interval [	frac34, 1], another length 	frac14. P = 	frac14 + 	frac14 = 	frac12 . Takeaway. A single uniform cut turns a ratio condition into an interval length. Requiring "longer \ge k tim…

One cut, how lopsided are the two pieces?

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