In recent years, software development models have undergone changes. In order to meet user needs and functional changes, open-source software continuously improves its software quality through successive releases. Due to the iterative development process of open-source software, open-source software testing also requires continuous learning to understand the changes in the software. Therefore, the fault detection process of open-source software involves a learning process. Additionally, the complexity and uncertainty of the open-source software development process also lead to stochastically introduced faults when troubleshooting in the open-source software debugging process. Considering the phenomenon of learning factors and the random introduction of faults during the testing process of open-source software, this paper proposes a reliability modeling method for open-source software that considers learning factors and the random introduction of faults. Least square estimation and maximal likelihood estimation are used to determine the model parameters. Four fault data sets from Apache open-source software projects are used to compare the model performances. Experimental results indicate that the proposed model is superior to other models. The proposed model can accurately predict the number of remaining faults in the open-source software and be used for actual open-source software reliability evaluation.