One cut, longer piece at least four times

Question

A wire of length 1 is cut at a single uniformly random point, making two pieces. What is the probability that the longer piece is at least 4 times the length of the shorter piece?

Accepted Answer

Let the cut be at U \sim U(0, 1), giving pieces U and 1 - U. For U \le 	frac12 (shorter U, longer 1 - U): 1 - U \ge 4U \quad\Longrightarrow\quad U \le 	frac15 . That interval has length 	frac15, and by symmetry the case U 	frac12 adds [	frac45, 1], another 	frac15. P = 	frac15 + 	frac15 = 	frac25 . Takeaway. The pattern 	frac{2}{k+1} shrinks as k grows: extreme lopsidedness needs the cut very near an end. With k = 4 the probability is 	frac25.

One cut, longer piece at least four times

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