This textbook aims to bridge a longstanding gap by presenting a rigorous yet physics-centered approach to probability theory and its applications. Despite its foundational role across nearly all areas of physics, many students exhibit notable technical and conceptual gaps in even the most basic aspects of probability. Surprisingly, these issues can persist even among those specializing in statistical mechanics. According to the authors, these shortcomings are largely a consequence of how probability is taught within the physics curriculum. Typically, core concepts are introduced early in a fragmented way, often within laboratory courses, and receive only cursory treatment in statistical mechanics classes. In contrast, more comprehensive and mathematically rigorous instruction is usually reserved for advanced graduate-level courses, which often lack a strong connection to physical intuition. This book addresses that imbalance by offering a cohesive and accessible framework tailored to the needs of physics students.