Animals possess complex innate computational capabilities essential for their survival, such as recognizing poisonous food or walking right after birth. However, specifying the precise wiring diagram for neuronal circuits to achieve these behaviors is too complex to be compressed through the "genomic bottleneck" [3]. Experimental data on cortical microcircuits [1][2] suggest that the genetic code determines synaptic connection probabilities between neurons based on their genetic types and spatial distances. We propose that these connection probabilities, rather than individual synaptic weights, are sufficient to encode innate computational capabilities in biologically plausible neural networks. We introduce the Probabilistic Skeleton (PS): a new mathematical framework that specifies a distribution of networks from synaptic connection probabilities, parametrized by neuron type and the distance between somata. Our results demonstrate that the PS framework can induce fundamental computing capabilities in networks of excitatory and inhibitory spiking neurons. Specifically, we tested the PS ability to recognize particular spatiotemporal stimuli, control motor functions, and generate patterns. Furthermore, our work shows that a substantial number of different neuron types is essential for this method to be effective, offering an explanation for the brain's employment of many more neuron types than previously considered in neural network models. Overall, our work provides evidence that the innate computational capabilities of the brain's neural networks can be programmed through a low-dimensional genetic code specifying synaptic connection probabilities. This approach circumvents the need for extensive learning processes and large datasets, offering new insights into the brain's efficiency and robustness.