For any fixed \(n\) and \(e > 0\), given a sufficiently long sequence of events in a probability space all of measure at least \(e\), some \(n\) of them will have a common intersection. This follows from the inclusion-exclusion principle. A more subtle pattern: for any \(0 < p < q < 1\), we can't find events \(A_i\) and \(B_i\) so that the measure of \(A_i\) intersected \(B_j\) is less than \(p\) and of \(A_j\) intersected \(B_i\) is greater than \(q\) for all \(1 < i < j < n\), assuming \(n\) is sufficiently large. This is closely connected to a fundamental model-theoretic property of probability algebras called stability. We will discuss these and more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory (no prior knowledge is expected—all of these will be introduced).