"What are the fundamental principles of probability theory, and how do they apply to statistical analysis? Can you elaborate on the concept of probability distributions and their relevance in modeling random phenomena? Furthermore, what methods are commonly used to calculate probabilities and expected values in statistical scenarios? Additionally, could you explain the distinction between discrete and continuous probability distributions, providing examples of each? Lastly, how do statistical techniques such as hypothesis testing and confidence intervals aid in making informed decisions based on data analysis?"