Why catastrophic claims justify coverage despite low frequency
Most insurance policies never pay a claim. Auto insurance customers in low-accident demographics might go 20+ years without filing. Home insurance holders might have zero claims in a 30-year mortgage. The modal outcome is 'nothing happens, you paid premiums for nothing', and this is correct.
Insurance economics relies on the fat tail: rare but devastating events that would bankrupt individuals. A house fire or car total loss costs $300,000. A liability judgment costs millions. These low-probability, high-consequence events justify the cost of coverage even when the statistical expected value is negative (premiums exceed average payout).
The asymmetric bet that insurance protects against
Insurance is backward from investment logic. You pay a premium expecting not to break even (insurers are profitable). The value comes not from financial return, but from converting a small probable cost (premiums) into protection against an impossible-to-absorb catastrophe.
Self-insuring works only for individuals rich enough to absorb worst-case loss without hardship. A billionaire might skip car insurance because $100k deductibles are inconsequential. A middle-class household cannot absorb a $500k liability judgment or total loss without insurance shifting that risk.