> Shouldn't that read: "" More specifically, the sum of all expected future premiums that all patients pay should be slightly more than the sum of all expected future claims for all patients ""
No, it shouldn't. I mean, the thing you wrote is a true statement too, because of the distributive property of summation, but the point is that insurance does work on a per-plan level. If any individual's expected future premiums are less than their expected future payouts, then it's not insurance.
There's a common misconception that insurance is about pooling risk between patients, which is not really the case - it's true at an accounting level, but only as a secondary effect. The fundamental purpose of insurance is to smooth risk across states of the world, which is a completely different thing.
Pretend for the moment that spontaneous combustion of humans exists, can be predicted with some understood probability model, and is an insurable event. In that case, "spontaneous combustion insurance" is not about smoothing risk between Alice and Bob - it's about smoothing risk between universe A (in which Bob spontaneously combusts tomorrow) and universe B (in which Bob does spontaneously combust).
Spontaneous combustion is an easy example to use illustratively because it's a binary event, terminal (you can only combust once), and intuitively independent (by definition of spontaneity, Bob's combustion does not impact Alice's chances of combustion). This is a simplified version of the general insurance model, however, which is capable of handling non-binary events, non-terminal events, and interdependent events. In all of those cases, the risk smoothing happens between states-of-the-world, not between individuals.
> If an insurer finds out that you will develop a certain type of cancer for which median treatment cost is 1M USD and lifetime support costs will be 30K USD/year, they should not be allowed to increase your premium accordingly.
That's a value judgement that you're free to make, but at that point, we're no longer talking about insurance in anything but name.