The whole reason insurance exists is because of differences in risk tolerance. What is a huge risk for me, such as a fire destroying my house, is a relatively small risk for an insurance company that is insuring against fires across the entire state. What I pay the insurance company for is to assume part of that risk.
Consider homeowners insurance, and more specifically fire insurance, in this admittedly contrived example. Suppose that in the next year there's a 1/1000 chance of a fire that will cause damage that will cost $100k to repair. That has an expected value of $100. Well, since $100k is a lot of money to me, I'd rather pay someone $200 than take a bet with an expected cost of $100, even though paying $200 has a negative expected value. That means I am risk averse for potential gains and losses on the order of $100k, and would rather take the more certain side of a bet, even if it means it has a lower expected value.
Take another example. Suppose I'm worried about losing or breaking my cell phone over the next year, and it would cost $500 to replace. AT&T charges $6.99/month for insurance on the phone. Over the course of a year that's about $84. And furthermore suppose there's a 1/20 chance that I'll lose/break/etc my phone during that year. Without insurance, the expected value of the loss is $25. Unlike the $100k example, $500 isn't that big a deal to me, so the insurance is a horrible deal for me, because I'm risk neutral for a $500 loss.
Of course, real life is more complicated. Homeowner's insurance protects against risks other than fire. Risks to the insurance company can be correlated - something on the order of the 1906 SF fire is a large risk, even to an insurance company, which is why there is reinsurance. There are deductibles that change the pricing. But still, as a simple example, that's how insurance works.