Why certain probability distributions do offer stochastical insight and others do not

Some standard probability distributions (like the binomial or Poisson) are especially attractive for teaching. They can be decomposed into smaller "moduls". These moduls are much easier to handle, a check of them could be a statistical test, but a judgement by mere reflection of the situation is also possible. It is argued that the process of decomposing into moduls and synthesizing them to the original situation enables insight into the stochastic structure of the standard situations.