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For a broad class of input-output maps, arguments based on the coding theorem from algorithmic information theory (AIT) predict that simple (low Kolmogorov complexity) outputs are exponentially more likely to occur upon uniform random sampling of inputs than complex outputs are. Here, we derive probability bounds that are based on the complexities of the inputs as well as the outputs, rather than just on the complexities of the outputs. The more that outputs deviate from the coding theorem bound, the lower the complexity of their inputs. Since the number of low complexity inputs is limited, this behaviour leads to an effective lower bound on the probability. Our new bounds are tested for an RNA sequence to structure map, a finite state transducer and a perceptron. The success of these new methods opens avenues for AIT to be more widely used.