A natural introduction to probability theory

In this introduction to probability theory, we deviate from the route usually taken. We do not take the axioms of probability as our starting point, but re-discover these along the way. First, we discuss discrete probability, with only probability mass functions on countable spaces at our disposal. Within this framework, we can already discuss random walk, weak laws of large numbers and a first central limit theorem. After that, we extensively treat continuous probability, in full rigour, using only first year calculus. Then we discuss infinitely many repetitions, including strong laws of large numbers and branching processes. After that, we introduce weak convergence and prove the central limit theorem. Finally we motivate why a further study would require measure theory, this being the perfect motivation to study measure theory. The theory is illustrated with many original and surprising examples.