A new class of solutions of the ultimate extinction probability for a neutron fission chain have been derived from Galton-Watson processes, which allows for a significant simplification in their derivation with respect to the conventional probability balance approach. These solutions are not based on the one-group diffusion hypothesis, depend only on the effective multiplication value and allow to take into consideration a large variety of discrete fission neutron distributions. They have been compared to other solutions, derived from probability balance equations, and they are qualitatively in good agreement. Additionally, the polynomial form of a model based on the geometric distribution presented here, has allowed for the calculation of the alive neutrons probability distribution that has been compared, for different generation numbers, with criticality benchmarks’ distributions estimated via a neutron Monte Carlo code. The comparisons have shown good agreement for generation numbers larger than 1.