1  What is probability?

Probability is the most important concept in modern science, especially as nobody has the slightest notion of what it means. —— B. Russell

What is probability? We all talk about probabilities in everyday life, but mostly in vague languages. This course is to introduce probability as a logical framework for quantifying uncertainty and randomness.

Mathematics is the logic of certainty; probability is the logic of uncertainty. —— J. Blitzstein

The earliest development of probability is rooted in gambling. The famous Monte Carlo method in statistics, invented by Stanislaw Ulam in the late 1940s, takes its name from the Monte Carlo Casino in Monaco, where Ulam’s uncle went to gamble.

Today, probability theory has been applied to almost every field of human knowledge. It is the foundation of statistics, machine learning, and artificial intelligence. It also plays a crucial role in everyday decision-making, from stock investments to effective strategies to combat an infectious disease.

The first formal definition of probability is often attributed to Pierre-Simon Laplace in the 18th century. In his work Theorie analytique des probabilites, published in 1812,

The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.

We will soon discover that this definition is obsolete. We start the journey of modern probability theory by introducing the basic concepts of events and sample spaces.