18.4 Summary

Phew! We are at the end of an intimidatingly long, albeit extremely essential chapter. Although we’ve talked about the mathematical details of probability for a couple of dozen pages, the most important takeaway can be summarized in a sentence: probability theory extends our reasoning toolkit by handling uncertainty. Instead of measuring the truthiness of a proposition on a true-or-false binary scale, it opens up a spectrum between 0 and 1, where 0 represents (almost) impossible, and 1 represents (almost) certain.

Mathematically speaking, probabilistic models are defined by probability measures and spaces, that is, structures of the form (Ω,Σ,P), where Ω is the set of possible elementary outcomes, Σ is the collection of events, and P is a probability measure, satisfying

1. P(Ω) = 1
2. and P(∪_n=1^∞−A_n) = ∑ _n=1^∞P(A_n) for all mutually disjoint A_n ∈ Σ,

which are called the Kolmogorov axioms. Thinking...