Lesson 3

Events \(A_{1}, \dotsc, A_{n}\) are called mutually independent if the probability of any of these events does not change when any subset of the remaining events occurs. This is equivalent to the following condition: for any subset of events, the probability of all of them occurring simultaneously is equal to the product of their probabilities. Formally, for any \(\varnothing \neq I \subseteq [n]\) it holds that \[\Pr\left[\bigcap_{i \in I}A_{i}\right] = \prod_{i \in I}\Pr[A_{i}] \ .\] Events \(A_{1}, \dotsc, A_{n}\) are called \(k\)-independent (or \(k\)-way independent) if any \(k\) of them are mutually independent. In particular, for \(k=2\), they are called pairwise independent.

It is easy to see that \(k\)-independence is a weaker property than mutual independence.