  We introduce a finite version of free probability and show the link between
recent results using polynomial convolutions and the traditional theory of free
probability. One tool for accomplishing this is a seemingly new transformation
that allows one to reduce computations in our new theory to computations using
classically independent random variables. We then explore the idea of finite
freeness and its implications. Lastly, we show applications of the new theory
by deriving the finite versions of some well-known free distributions and then
proving their associated limit laws directly. In the process, we gain a number
of insights into the behavior of convolutions in traditional free probability
that seem to get lost when the operators being convolved are no longer finite.
This version contains the original preprint from 2016 as well as an extra
section (Section 5.2) where we use finite freeness to prove majorization
relations on certain convolution.
