I'd rephrase that first sentence of yours: You buy insurance for rare events. For things with a probability of less than one (over a reasonable time period). If something is likely to happen twice over, eg, a five-year period, you might as well just account for it in your budget and make sure you have enough cashflow to cover it.
But insurance can also serve as a protection against bad luck. For example, there might only be a 20% chance you break the back cover glass of the iPhone X over a two-year period. But when it happens to you twice, you might end up paying more for repairs than the initial purchase price. Of course, if that 20% number is right, over a ten-year period (which might mean five iPhones), you'd only have to pay for a replacement once. Bad luck might really only be bad luck, but it can also be an initial underestimation of the likelihood of an event.
Paying for an insurance is like putting an upper limit on what you want to spend on something. Such a spending cap might not be rational but it is a way our brain simplifies complex tasks, by compartmentalising. In a similar sense it is a hedge against an estimation error on your side (which would be that your 'accident' rate is close to the average or lower).
In a similar vein there is the peace-of-mind argument. Having an insurance might make you more content ('happier') than knowing over your lifetime, you'll end up with more money by not buying certain types of insurance. The mind isn't always rational and sometimes it is easier to be rational about your irrationalities than trying to change the irrationalities.