Cutting random pieces of string

From a string of length \(1 \text{ m},\) we choose a length uniformly at random between \(0 \text{ m}\) and \(1 \text{ m},\) and cut as many of these lengths as possible from the string.

What is the expected length of the remaining string (in meters)?


Bonus: If we have a random number \(a\) chosen with uniform distribution over the interval \(\big(0,\frac{1}{n}\big],\) where \(n\in\mathbb{N}\), what's the expected value of the remainder \(1\bmod a\) for a particular \(n?\)

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