  We present a generalized stochastic Cantor set by means of a simple {\it cut
and delete process} and discuss the self-similar properties of the arising
geometric structure. To increase the flexibility of the model, two free
parameters, $m$ and $b$, are introduced which tune the relative strength of the
two processes and the degree of randomness respectively. In doing so, we have
identified a new set with a wide spectrum of subsets produced by tuning either
$m$ or $b$. Measuring the size of the resulting set in terms of fractal
dimension, we show that the fractal dimension increases with increasing order
and reaches its maximum value when the randomness is completely ceased.
