Sure, but mixture models and mutually exclusive events are less common to work with than just "models." Obviously there are times when you add probabilities together, but usually you don't. Even when you're writing a mixture model, or doing bayes nets, most of the work is done with log probabilities in the various branches, and only retransformed into linear space when the branches are combined.
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Mixture models are surprisingly common and addition/subtraction are not limited to mutually exclusive events. In such cases one goes the route of inclusion/exclusion formula.
Summation is so fundamental to probabilities that it is how probabilities are even defined axiomatically in the Kolmogorov system. https://en.wikipedia.org/wiki/Sigma_additivity
I think what you meant was that in computation one often needs to multiply probabilities and in such cases it is helpful to work in the log semiring https://en.wikipedia.org/wiki/Log_semiring
> Obviously there are times when you add probabilities together, but usually you don't.
Maybe you don’t but it’s hardly unusual. Integrating out nuisance parameters from a posterior distribution is adding probabilities. Monte Carlo methods are based on adding probabilities.
>Sure, but mixture models and mutually exclusive events are less common to work with than just "models."
What do you mean by this? A mixture model is just a model.
Also, how can you do absolutely anything without "working with mutually exclusive events"? You need mutual exclusivity to support the concept of a decision or a hypothesis.