[babble-on]

"My bad" as an expression comes from baseball, specifically the game of "catch". Two people are playing catch. One of them makes a bad throw. The other person has to turn around and chase down the ball. The person who threw it will say "My bad" by way of apology. What is good about this expression is its succinctness, ideal for informal setting such as a game of catch, when it is nevertheless necessary to establish quickly who it was who screwed up.

[11th Earl of Mar]

Please accept my apologies for the fact that I short-hopped the ball to you, friend

I would simply say, "My fault."