"Expected utility is that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average, where the weights are the agent's estimate of the probability of each state." -Daniel Bernoulli-

My assumption is that the demand for health insurance is that people are happier (larger utility) paying a certain small amount each month to protect against the risk of a uncertain large loss that might occur. Thus, people are able to calculate their expected utility in that they will statistically have a large financial loss in the future. An example would be if my family had a history of heart disease. I would expect I would have heart disease in my future. Knowing that I would want to protect as well as I could against having heart disease this includes a well balanced diet. However I still want health insurance because the probability that I will have heart disease is large. I would want to pay a couple hundred dollars a month for the rest of my life instead of having a medical bill of several hundreds of thousands of dollars. In conclusion people demand health insurance because they want to avoid a large financial loss in their future.

References:
A quote from Daniel Bernoulli

"The Deman for Insurance: Expected Utility Theory from a Gain Perspective"
John A. Nyman
University of Minnesota