From Flajolet/Segdewick's Analytical Combinatorics (p. 53) I have learned that 97% of the binary sequences of length 200 contain runs longer than 5. The other way around: 97% of the sequences contain no run longer than 11.
Let $a(n)$ be the minimal number $a$ such that 97% of the binary sequences of length n contain no run longer than $a$. Example: $a(200) = 11$
What is the limiting behaviour of $a(n)$?
(in Big-O-notation).
A related question is:
What is the limiting behaviour of the maximum $max(n)$?
Example. $max(200) = 7$.