-> In the problem, the sample space comes divided into mutually disjoint portions.

-> Using the definition of probability, we can compute joint distribution from the a-priori distributions.

-> Using the definition of conditional probability, we can flip around the inference direction.

I usually find that all problems (Naive-Bayes spam, cancer) etc. all show this pattern.

It works especially well with the examples, like Yudkowsy's breast cancer screeening example.

Jaynes argues here (pg 221) that examples were Venn diagrams apply are special cases. According to Jaynes, Bayes theorem is more general than that.

  world = get_world()
  for experiment in experiments():
    world = combine(world, experiment())