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This course offers a rigorous introduction to the mathematical foundations of probability and statistics, emphasizing applications across the natural, physical, and social sciences. Students will explore the axioms of probability theory, combinatorial analysis, and fundamental concepts such as sample spaces and events. Key topics include conditional probability, Bayes' theorem, and the law of total probability, providing a framework for understanding complex probabilistic scenarios.The curriculum delves into discrete and continuous random variables, examining distributions such as binomial, geometric, Poisson, exponential, uniform, and normal. Students will learn to compute expectations, variances, and apply the law of the unconscious statistician for function transformations. The course also covers joint, marginal, and conditional distributions, covariance, and independence, with an introduction to moment-generating functions.A significant focus is placed on the Central Limit Theorem and its implications for statistical inference. Students will engage with real-world applications, enhancing their analytical skills and preparing them for advanced studies in statistics, data science, and related fields. A solid understanding of Calculus II is required, and familiarity with multivariable calculus is beneficial for topics involving joint continuous distributions.Students with minimal calculus backgrounds can still benefit from many early topics and especially discrete distributions. One can still learn the underlying theory of probability without a background in Calculus

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