I've searched on Google and this site, but haven't found an answer.
When Rey and Finn were running towards the first ship, the TIE fighters blew it up before they got to it. Rey exclaims the "The garbage will do!" and they both make a bee line for the Falcon. The TIE's don't shoot it.
They spend another minute or so getting used to the guns and pilot seat, and again the TIE's don't shoot. It's stationary at that point and one of the larger ships in the area so it was an easy target.
Only when they finally (clumsily) start taking off do the TIE's shoot, and somehow completely miss, despite their pin-point accuracy on the first ship they blew up.
Why (1) did they not shoot while Rey and Finn were running towards it and warming it up and (2) did they miss while it was getting off the ground?