  We obtain an index of the complexity of a random sequence by allowing the
role of the measure in classical probability theory to be played by a function
we call the generating mechanism. Typically, this generating mechanism will be
a finite automata. We generate a set of biased sequences by applying a finite
state automata with a specified number, $m$, of states to the set of all binary
sequences. Thus we can index the complexity of our random sequence by the
number of states of the automata. We detail optimal algorithms to predict
sequences generated in this way.
