Sure! To quote Scott's own TL/DR: Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try

Sure! To quote Scott's own TL/DR: Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call "probabilities" can be negative numbers. That's it. So in QM you can model events whose probabilities 'interfere' with each other by canceling each other out (eg independent events A and B have 'probabilities' -20% and +20% respectively, but you want the 'probability' of either A or B occurring to be 0%), and do all other sorts of weird stuff. Now why would you want to do such a thing is a whole different matter. But at least this should get you started.