Mathematical modeling has long been a valuable tool for studying the dynamics of infectious disease outbreaks. Among the most widely used models is the SIR (Susceptible-Infected-Removed) framework, which classifies a population into these three classes and uses analytical tools to study the dynamics of each classes. This article provides a comprehensive study of the deterministic SIR model as well as a stochastic SIR model that accounts for randomness. The deterministic model, dating back to the pioneering work of Kermack and McKendrick in 1927, uses a system of differential equations to track the time evolution of the S, I and R population groups. A key concept is the basic reproduction number R0, which determines if an outbreak will occur or die out and provides a tool to estimate the final size of the epidemic. While the deterministic model provides a big-picture view, the stochastic SIR model captures the randomness in real-world problem. Formulated as a continuous-time Markov chain, it includes transition probability at each time step. Interestingly, even when R0>1, the stochastic model shows there is still a chance that only a minor outbreak happen. The article then dives into statistical methods - namely, Bayesian inference and data augmentation techniques - to estimate the model parameters such as transmission and recovery rates from actual outbreak data. Our main work explores a framework for sequential daily prediction of new cases and the final size of the outbreak. Both simulated datasets or real world datasets like Ebola and COVID-19 are analyzed using this methodology. We include the model's performance, predictions, and limitations of this simplified stochastic SIR model.