One cut, a very short scrap

Question

A string of length 1 is cut at one point chosen uniformly at random, giving two pieces. What is the probability that the shorter piece is less than 	frac{1}{10} of the whole string?

Accepted Answer

Let the cut be at U \sim U(0, 1); the shorter piece has length \min(U, 1 - U). The shorter piece is under 	frac{1}{10} exactly when the cut lands within 	frac{1}{10} of an end: U < 	frac{1}{10} \quad	ext{or}\quad U 	frac{9}{10} . Each interval has length 	frac{1}{10}, so P = 	frac{1}{10} + 	frac{1}{10} = 	frac15 . Takeaway. "Shorter piece below c" is simply the cut landing in the outer c at either end, probability 2c. Here 2 \cdot 	frac{1}{10} = 	frac15.

One cut, a very short scrap

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