What the law of large numbers actually promises, and what it does not. As a pool grows, the number of losses it experiences gets closer to the expected number in percentage terms, even while the absolute variation gets larger. That is the whole of the guarantee, and it is a statement about the insurer's books rather than about any policyholder. A person in a pool of ten million faces exactly the probability of loss they faced in a pool of ten. Consumer writing frequently gets this backwards and describes pooling as though it spreads a person's own risk thinner; it does not. What it spreads is the insurer's uncertainty about the total.
Independence is the assumption that carries the whole structure. The arithmetic in the section below only works if one member's loss tells you nothing about another's. Where a single event can strike every member of the pool in the same instant, the losses arrive together and the average is no longer a useful prediction. That is the structural reason catastrophe perils sit outside standard property coverage rather than being merely expensive: earthquake and flood damage whole regions at once. The market's answers to that problem are all attempts to restore independence or to find a bigger balance sheet: reinsurance moves part of the exposure to a global carrier whose own book is spread across unrelated geographies, and where even that is not enough, the pool is created by statute.
What makes a risk poolable in practice. Insurers write coverage where losses are accidental from the insured's point of view rather than chosen, where the loss and its cause can be identified and measured, where the chance of loss can be estimated from enough past experience to price it, and where enough similar exposures exist to make the average meaningful. Those conditions describe most of what is sold and explain most of what is not. A loss the insured controls is a problem addressed by cost-sharing design; a loss the buyer knows about and the insurer does not is addressed by underwriting and enrollment rules; a loss that hits everyone at once is addressed, or not, by the arrangements above.
The named pools are where policy shows through. Insurance regulators define an assigned risk plan as "a governmental pool established to write business declined by carriers in the standard insurance market," and a FAIR plan, from Fair Access to Insurance Requirements, as a state pool "designed to provide insurance to property owners who are unable to obtain property insurance through conventional means." Texas runs an automobile version of the first: its insurance regulator explains that a driver who cannot find a company willing to sell them a policy can get basic coverage through the Texas Automobile Insurance Plan Association after two companies have turned them down, that the association sells liability, personal injury protection and uninsured or underinsured motorist coverage but not collision or comprehensive coverage or limits above the state minimum, that it is more expensive than ordinary coverage, and that after three years without a ticket or accident the insurer must offer a cheaper policy outside it. That last provision is the pool behaving as designed: a residual market is meant to be a way back into the standard one, not a destination.
California built the same kind of institution for a peril rather than for a population. Faced with insurers unwilling to carry earthquake exposure, the legislature created the California Earthquake Authority by statute in 1995 and required residential property insurers to offer earthquake coverage. The detail belongs on the earthquake insurance page; the structural point belongs here. Where private pooling of a correlated risk does not happen, the alternatives are a statutory pool, a federal program, or nobody carrying the risk at all.
The investing parallel is exact, and so is its failure mode. Holding many uncorrelated positions reduces the variability of a portfolio's return the same way holding many uncorrelated policies reduces the variability of an insurer's loss ratio, and in both settings the reduction disappears as correlation rises. A portfolio of forty holdings that all fall together in a panic is, at that moment, one holding, and a pool of forty thousand houses on one fault line is one house.
