I would like to be able to illustrate why additivity is a natural property to assume for probability measures. While it is relatively simple to give a short hand waving intuitive analogy to volume or mass I feel this is somewhat unsatisfying, not least since it does not really illustrate how you would use it. What seems most reasonable to me is to show what amounts to an application of the law of total probability. The problem is that all examples I can think of rely on conditioning or independence, and in the standard measure theoretic (Kolmogorov axiom) formalism the topic of additivity is, by necessity, raised before one can get to independence or conditioning. I would prefer to be able to give at least one motivating example of additivity before going on to the next topic. The best substitute I have been able to think of is to take a proof of some basic proposition for probability measures that invokes additivity (such as monotonicity with respect to ordering of events) and specialise it to some concrete events. I worry, though, that students with little mathematical background (the ones I'm teaching this semester have not even seen basic calculus) would find such arguments overly contrived or circuitous. Note that I realise that one could approach the subject completely without considering assignment of probabilities as something reifed as a measure. That being said, since the course is also to cover basic statistics I feel like this is more or less necessary considering how statistics requires one to deal of multiple alternative (sets of) measures corresponding to different hypotheses.
EDIT: In hindsight I think I made the mistake of trying to anticipate responses, thus making the question much more convoluted than necessary. Let me therefore try to state it more concretely.
Is it possible to give an example of how additivity/disjointness can be used to analyse a complex event by partitioning it into simpler parts, before conditioning and/or independence has been introduced. Since the idea is to illustrate how additivity/disjointness encode something important/natural I want an example with some mathematical content (i.e. I would like to avoid a pure appeal to intuition or prototypical examples) but not something purely mathematical (i.e. not only derive additional nice properties measures will have).
The example I ended up using was, if I remember correctly, one that was also hinted at here. Namely that it allows one to motivate the 1/6 probability for a die roll from symmetry of the die. I don't think this worked particularly well though, since I think many people did not even realise this probability assignment was something that had to be justified.
Failing this, can one give some other example (not in terms of partitioning a complex event) achieving the same goal?