The probability of a particle being at speed $v$ or its corresponding energy $E$ via the MB Distribution $f(v)$ and $f(E)$ is
$$f(v)\cdot dv=f(E)\cdot dE$$
To make this equation valid, either the $dE$ must correspond to a fixed $dv$ or vice versa according to $E = \frac{1}{2}mv^2$
$$f(v)\cdot dv = f(E)\cdot mv\cdot dv$$
$$or$$
$$f(v)\cdot \frac{dE}{\sqrt{2mE}}= f(E)\cdot dE$$
Because of these two versions, there are 2 ways to get back to the MB Distribution which is a probability *density*.

1. Divide $f(v)dv$ by a fixed $dv$ and therefore divide $f(E)dE$ by the corresponding varying $dE = mv\cdot dv$
2. Divide the $f(E)dE$ by a fixed $dE$ and therefore divide $f(v)dv$ by the corresponding varying $dv = \frac{dE}{\sqrt{2mE}}$

For *the same* $v$, the emerged $f(v)$ from method 1 gives a different probability density than the emerged $f(v)$ from method 2. Idem for $f(E)$ for the same $E$.

So if I have $f(v)$ in front of me without knowing whether it was deduced from method **1** or **2**, how do I know whether I should multiply it by a fixed $dv$ or a varying $dv = \frac{dE}{\sqrt{2mE}}$ to get the correct probabilities again?