The Johnson-Cook (J-C) model, including a constitutive model and a failure criterion, is an empirical model that is widely used in various fields to implement large deformation and damage of materials. Experimental methods are selected to determine the J-C model parameters considering the strain rate and condition where the material is exposed. This study deals with a uniaxial tensile test performed to obtain the stress–strain response of a material under quasi-static strain rate conditions. However, a low coefficient of determination may appear in a linear regression that determines the strain rate sensitivity parameters. Furthermore, in terms of efficiency, it requires a large amount of test data to obtain a set of the J-C model parameters. The novelty of this study lies in the adoption of a probabilistic approach to identify the J-C model parameters, which resolves the limitations of the existing method while simultaneously improving efficiency and accuracy. Namely, the efficiency is secured since all the J-C model parameters derived from each uniaxial tensile test response are used to estimate the probability model without the dissipation of the test data. Moreover, the proposed method is also efficient since it is capable of sampling as many of the J-C model parameters as necessary from the estimated probability model utilizing D-vine. Lastly, the inherent variability of material properties is accurately modeled by estimating the joint probability distribution as copulas, which can reflect the correlation between the J-C model parameters. The implemented statistical strategies are divided into two stages: (1) estimating the marginal probability distribution through the maximum likelihood estimate (MLE) and (2) the joint probability distribution through the Bayesian method. The two numerical examples adopted to validate the proposed method demonstrate that the proposed method better matches the experimental data rather than the existing deterministic method or the probabilistic method whose correlation has not been considered.