# How do incompressible strings and random strings share the same properties?

I came across the following theorem in Sipser's about incompressible strings.

Let $\;f$ be some computable function which holds for almost all strings. The for any $b > 0$, the property $\;f$ is false only on finitely many strings that are incompressible by $b\;$.

where "almost all strings" means that as $n$ grows the fraction of strings of length $n$ for which $f$ is false approaches $0$.

The book proves the above theorem and says that if we select some sufficiently long string at random, it's likely that property $f$ would hold true for it. Thus incompressible strings and random strings share this property.

Though I understand the proof for the theorem, I am unable to understand its meaning. How does this show there is some sort of relation between incompressible strings and random strings?

Also how does this theorem help us? Although I can generate random strings, there is no method to generate incompressible strings in general.

• what section is that? ps in some sense if you understand the proof then you understand the meaning. :) – vzn Nov 26 '15 at 20:38