A combinatorial problem related to the classical probability

TL;DR

This paper investigates the maximum number of pairwise independent events in a classical probability space, establishing bounds and conditions related to Hadamard matrices and intersecting families.

Contribution

It proves an upper bound for the maximum number of pairwise independent events and links the exact value to the existence of Hadamard matrices.

Findings

f(n) q n+1 if a Hadamard matrix of order n exists

f(n) q n+1 for all n, with bounds proven otherwise

Connection established between combinatorial families and probability independence

Abstract

In the classical probability model, let $f(n)$ be the maximum number of pairwise independent events for the sample space with $n$ sample points. The determination of $f(n)$ is equivalent to the problem of determining the maximum cardinality of specific intersecting families on the set $\{1,2,\ldots,n\}$ . We show that $f(n)\leq n+1$, and $f(n)=n+1$ if there exists a Hadamard matrix of order $n$.