Insurance is about trading money for utility. The assumption is that your utility curve in the lossy region is sublinear, i.e. U(-$1e6) << 1e6 x U(-$1), and that the probability of a large loss is low (e.g., 1e-6).
In that case, if you pay a guaranteed -$1, your expected utility loss is U(-$1). If you have a 1e-6 chance of losing $1e6, your expected utility loss is 1e-6 U(-$1e6) < U($1).
Thus, it makes sense to pay $1 to avoid the risk of losing $1e6.
Insurance which pays for high probability, low cost events (e.g., gas for your car, birth control pills) is indeed foolish.
> The assumption is that your utility curve in the lossy region is sublinear
Which makes lottery tickets all the more a bad idea. Not only is the expected return in dollars less than your investment, but thanks to the diminishing marginal utility of money your ten millionth dollar will be worth less than your ten thousandth. Lottery tickets are actually worse than their already crappy EV.
Well, some people hypothesize that your utility can be superlinear in the positive region. Utility = (gain or loss)^3, for example.
Of course, the stats prof says he buys lottery tickets because they are fun. I do something similar - even though the expected gain from video games is precisely $0, I still play them.
In that case, if you pay a guaranteed -$1, your expected utility loss is U(-$1). If you have a 1e-6 chance of losing $1e6, your expected utility loss is 1e-6 U(-$1e6) < U($1).
Thus, it makes sense to pay $1 to avoid the risk of losing $1e6.
Insurance which pays for high probability, low cost events (e.g., gas for your car, birth control pills) is indeed foolish.