Probabilistic insurance is a concept introduced by Kahneman and Tversky in their 1979 paper on prospect theory. Here is how it works.
You want to insure a property against damage. After inspecting the premium, you find it difficult to decide to pay for the insurance or leave the property uninsured. Now, you get an offer for a different product that has the following feature:
You spend half the premium but buy probabilistic insurance. In this case, you have a probability p, e.g. 50%, in which you, in case of damage, will pay the rest of the 50% and get fully covered, or the premium is reimbursed, and the damage goes uncovered.
For example, the scheme works in the first mode (pay the reminder and full coverage) on odd days of the month and the second mode (reimbursement and no coverage) on even days!
Intuitively unattractive
When Kahneman and Tversky asked this question to the students of Standford university, an overwhelming majority (80%) was against the insurance. People found it too tricky to leave insurance to luck or chances. But in reality, most insurances are probabilistic, whether you are aware or not. The insurer always leaves certain types of damages outside their scope. The investigators proved using the expected utility theory that probabilistic insurance is more valuable than a regular one.
Tversky, A.; Kahneman, D., Econometrica, 1979, 47(2), 263
