In particle physics their are several quantities (charge, baryon number, lepton number, momentum, and energy) which are conserved by all 4 fundamental interactions. Often in textbooks etc. their are questions asking whether a particular interaction can occur. e.g. $$A +B \rightarrow X+Y$$ I usually do such questions by seeing if any of the above conservations laws have been violated, and if they haven't stating that it can occur. But this got me thinking, is it in fact a necessary and sufficient condition for an interaction to have a non-zero probability of occurring that these conservations laws are satisfied? Or are their interactions that have a zero probability but do satisfy these conservations law? Please can you explain your answer.
As indicated in the comments, the totalitarian principle outrightly forbids this. However, it would be better if you could actually know at least one of the reasons why, which is a rather big one.
Suppose you take some some conservation laws into account and on that basis you want to create a particle interaction. From Noether's theorem, we know that there are some specific symmetries to be obeyed by the Lagrangian of the system in order for the conservation laws to hold true. But that in itself doesn't fix the Lagrangian at all. There are many possible particle interactions generated by many Lagrangians, and they all need not be necessarily obeyed.
A simple example is a massless scalar particle without any potential and mediating fermionic interactions, which is a perfectly valid model obeying all conservation laws. But it simply doesn't exist, or we would have observed a fifth fundamental force in nature, which would also be inverse-square.