  Let $\Theta$ be a finite alphabet. We consider a bundle of measure preserving
transformations $(T_{\theta})_{\theta \in \Theta}$ acting on a probability
space $(X,\mu)$, which are chosen randomly according to an ergodic stochastic
process $(\Xi,\nu,\sigma)$ with state space $\Theta$. This describes a
paradigmatic case of a random dynamical system (RDS). Considering a finite
partition $\mathcal{P}$ of $X$ we show that the conditional algorithmic
complexity of a random orbit $x, T_{\alpha_{0}}(x),T_{\alpha_{1}}\circ
T_{\alpha_{0}}(x),...$ in $X$ along a sequence $\alpha =
\alpha_{0}\alpha_{1}\alpha_{2}...$ in $\Xi$ equals almost surely the fiber
entropy of the RDS with respect to $\mathcal{P}$, whenever the latter is
ergodic. This extends a classical result of A. A. Brudno connecting algorithmic
complexity and entropy in deterministic dynamical systems.
