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We start by giving a definition of probability. It is easier to understand with an example. Imagine you have a fair coin. Obviously, the probability is 1/2. Now try some exercises. – 99% of books that have “introduction” and “probability” in the title Probability is defined as a measure. Distribution functions, expectations, and many others are defined using the Lebesgue integral. Most people who read an introduction to probability do not know what a Lebesgue integral is, so textbook authors avoid talking about it. It’s stupid, because it guarantees that you will have holes in your understanding of the basics. ...

I have a habit of regularly revisiting the advice pages I keep bookmarked. Most of this advice seems obvious to me, yet I still manage to occasionally forget to follow it. One of those pages is rather strange. It’s Terence Tao’s. His suggestions are divided into five categories, from primary school to postdoctoral research, gradually shifting from “do not focus on tests and exams” to “do not obsess over famous problems.” I’m not a professional mathematician, and at this point, I probably never will be. Still, whenever I reread these mostly irrelevant tips, I find myself energized, driven to work, and my mood instantly improves, more so than from any other advice. It always felt odd. ...