It is generally accepted knowledge that when eve ate from…

It is generally accepted knowledge that when eve ate from the tree of knowledge of good and evil, that was the original sin. However, the serpent/Satan was already in the garden before mankind was created, meaning him and 1/3 of the legion of angels had fallen. We have to assume that you cannot fall from heaven without committing sin. So my questions is this, did satan or eve actually commit the first sin?