IF so, how do they relate?
So far as I know Bernoulli's equation is not mathematics but something to do with the physics of air lift.
But there is Bernoulli's inequality: SEE HERE
Where we see that $(1+x)^n\ge 1+nx,~n\in\mathbb{Z},~n\ge 2~\&~x\ge -1$ now that is mathematics.
Here is binomial distribution $\displaystyle {(1 + x)^n} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right){x^k}} $.
Now what more is there to your question?
There is also a differential equation called Bernoulli's equation: $y' + P(x)y = Q(x)y^n$. Not related to binomial distribution that I know of. See
Differential Equations - Bernoulli Differential Equations
I think what Plato was reaching for in Physics is an equation in fluid dynamics. It is an extreme of fluid dynamics in that it deals with a smooth (no friction) irrotational (the fluid flow does not "twist") constant density linear (no vortices) flow of fluid, usually thought of as in a pipe. It's about as a simple concept as fluid flow gets. See here for the basics.
-Dan