I came to know that, in a random string, one expects the longest sequence of zeros to be roughly of length $\log n$. I want to be able to prove this. For this I need to know the probability that the longest sequence of zeros in a random $n$-bit string is of length $k$. Basically I need a good counting strategy to count the number of such $n$-bit strings i.e., where maximum run of zeros is of length $k$.
Can somebody give a hint on how to proceed? I don't want a full answer.