I make plenty of "mistakes" during a gig, though my definition of mistake is different from the audience's maybe. I'm thinking about the right note, of course, but also the rhythm, tempo, speed, attack, sustain, etc. and there are plenty of times I'll think while I'm playing "I didn't quite hit that the way I wanted."
Hitting the absolute wrong note or losing my place in a song--probably happens about once a gig.
There have been a few times when the band has started a song in the wrong key. We just laugh it off and start over and the audience never seems to mind.