Is there a definition of foosball random shots?
Do people track them somehow for fun as a "debt" as part of championship ladder?
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I assume you are asking about "official" foosball rules (ITSF)? Otherwise, there can be definitions just about everything because your local home games can make rules up.
In ITSF, there's no "random" shots, all shots that goes in, will count as a normal point (if it's shot legally). There's no point in tracking them any differently. How would one really define a random shot? It would depend entire on player skill, some player can reliably see a gap and place a 2-bar shot straight in or even bank it in, but another less skilled player might do that "randomly"... How would you determine that?
In high level gameplay, almost no shot "randomly" goes in. Even a quick pickup and attempt on goal from a loose ball is general high percentage because of proper angling and lightning fast aiming.