Negative expected dollar value is not the same as negative expected utility. Buying any sort of insurance has the former property (subject to some assumptions t

Negative expected dollar value is not the same as negative expected utility. Buying any sort of insurance has the former property (subject to some assumptions that may not actually be true, such as unlimited credit); it need not have the latter. Suppose, for instance, your total assets are $1M, and that you are extremely healthy but have a 0.1% chance each year of contracting a condition that will kill you rapidly (and agonizingly) unless it is treated, which can be done reliably and safely but costs $2M. (Yes, this is an artificial example.) Then the naive expected cost of your health situation is $2k/year. But if (a) you value your life at a lot more than $1M and (b) raising the money to pay for the treatment would impose a lifetime cost on you that's a lot more than $1M, then in terms of expected utility you would do better to pay for insurance against that condition, even if it costs you (say) $3k/year.