This paper provides the prediction of ultimate longitudinal strengths of the hull girders of a very large crude carrier considering probabilistic damage extent due to collision and grounding accidents based on IMO Guidelines (2003). The probabilistic density functions of damage extent are expressed as a function of non-dimensional damage variables. The accumulated probabilistic levels of 10%, 30%, 50%, and 70% are taken into account for the estimation of damage extent. The ultimate strengths have been calculated using the in-house software called Ultimate Moment Analysis of Damaged Ships which is based on the progressive collapse method, with a new convergence criterion of force vector equilibrium. Damage indices are provided for several probable heeling angles from $0^{circ}$ (sagging) to $180^{circ}$ (hogging) due to collision- and grounding-induced structural failures and consequent flooding of compartments. This paper proves from the residual strength analyses that the second moment of area of a damage section can be a reliable index for the estimation of the residual ultimate strength. A simple polynomial formula is also proposed based on minimum residual ultimate strengths.