Free probability is a subfield of mathematics at the intersection of operator algebras, complex analysis, probability, and combinatorics. It is used, among other things, to study the ``$n=\infty$'' case of various $n \times n$ random matrix models. A concept of central importance in free probability is \textit{free independence}, the ``noncommutative analogue'' of independence (of random variables) from classical probability. The goal of this talk is to develop a rigorous understanding of the throw-away clause in the previous sentence with an interesting mix of analysis and algebra. Time permitting, we may also discuss why classical independence and free independence are in a precise sense the only ``reasonable'' notions on independence.