One of our commenters has raised an interesting question: "Isn't risk just a fancier word for bad luck?"
Risk is about "the potential harm that may arise from some present process or from some future event," whereas luck is "an unknown and unpredictable phenomenon." The former has to do with probability, the latter with randomness.
By way of example:
Suppose I bet that you won't make a hole in one, blindfolded, at our local course. My risk is that you may, in fact, ace the hole. But the laws of probability dictate that there is a finite chance of this happening, and I can buy insurance to cover the eventuality. You, on the other hand, must depend on luck that you'll make the shot.
In the same way, health insurers know that there is a chance (a risk) that you will become seriously ill in the next year. But they also know, based on your application and their own statistics, the likelihood of this happening, and can arrive at a price that will cover that risk (premium). You may or may not become ill, but that's your luck, not the carrier's risk (which they've already managed).
Insurance is about spreading the risk; that is, assessing the likelihood that, out of a large group of people, one particular person will have a large claim. So if a person has the bad luck (we'll leave out behavioral choices for this example) to have, say, a heart attack, the company has already planned for that contingency.
Insurance is not about "spreading the luck:" there are other industries for that.