It depends on the level you are playing at and the type of shows you play IMO.
IF I was playing in a fingerstyle "blues gig", 3 sets a night , doing covers at the local dive bar to the people that just happen to be in the bar drinking anyway... I probalby woudn't be overly concerned about a backup, (but I personaly would likely bring one anyway.)
Playing the type of shows where people are coming to "see the show", not just drink and listen to whomever happens to be playing (BIG difference in type of show IMO), then it makes sense to have a backup. Particularly on a multi band bill where you are only allotted a specific amount of time to play. Unfortunatly, I've seen bands really screw over the Headlining Act, by monkeying around with malfunctioning gear and wasting show time.
That said... the stuff I do now is fairly agressive pick style playing, and yes.... strings break. They break on every bass I've ever owned and they are all VERY well mantained.
Having a backup is an absolute neccesity for me personaly. Not going to embarass myself by stopping a show to change a string.
Other issues where a back up has saved the bacon.
Broken strings.
Malfuctioning strap lock.
Screw stipped out of strap pin
Strap pin screw sheared off.
Broken nut. (o.k. I was playing guitar on that gig, but it's still relevent)
My guitarist had a Dunlop strap lock blow a retaining clip on stage, which dropped a Gibson Flying V to the floor, and sheared off a tuner.
Now, if you are just gigging casualy, the law of averages might take a long time to catch up with you. Question is... do ya feel lucky?
EDIT TO ADD: Of course , a roll of duct tape on hand is always a good thing too.
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