For almost all infinite binary sequences of Bernoulli trials $(p,q)$ the frequency of blocks of length $k\left(N\right)$ in the first $N$ terms tends asymptotically to the probability of the blocks, if $k\left(N\right)$ increases like ${\log }_{\frac{1}{p}}N-{\log }_{\frac{1}{p}}N-\psi \left(N\right)$ (for $p\le q$) where $\psi \left(N\right)$ tends to $+\infty$. This generalizes a result due to P. Flajolet, P. Kirschenhofer and R.F. Tichy concerning the case $p=q=\frac{1}{2}$.