270 D, KAHNEMAN AND A. TVERSKY occurs on an even day of the month, your insurance payment is refunded and your losses are not covered Recall that the premium for full coverage is such that you find this insurance barely worth its cost. Under these circumstances, would you purchase probabilistic insurance Y N=9520][80]* Although Problem 9 may appear contrived, it is worth noting that probabilistic insurance represents many forms of protective action where one pays a certain cost to reduce the probability of an undesirable event-without eliminating it altogether. The installation of a burglar alarm, the replacement of old tires, and the decision to stop smoking can all be viewed as probabilistic insurance The responses to Problem 9 and to several other variants of the same question indicate that probabilistic insurance is generally unattractive. Apparently, reduc ing the probability of a loss from p to p/2 is less valuable than reducing the probability of that loss from p/2 to 0 In contrast to these data, expected utility theory(with a concave u)implies that probabilistic insurance is superior to regular insurance. That is, if at asset position w one is just willing to pay a premium y to insure against a probability p of losing x, then one should definitely be willing to pay a smaller premium ry to reduce the probability of losing x from p to(1-r)p, 0<r<1. Formally, if one is indifferent between (w-x, P;w, 1-p)and (w-y, then one should prefer probabilistic insurance(w-x, (1-r)P;w-y, rp; w-ry, 1-p)over regular insurance(w-y) To prove this proposition, we show that p(w-x)+(1-p)u(w)=u(w-y) implies (1-r)pu(w-x)+rpu(w-y)+(1-p)u(w-ry) Without loss of generality, we can set u(w-x)=0 and u(w)=1. Hence u(w y)=1-p, and we wish to show that r(1-p)+(1-p)u(w-ry)>1-por(w-ry)>1- which holds if and only if u is concave This is a rather puzzling consequence of the risk aversion hypothesis of utility theory, because probabilistic insurance appears intuitively riskier than regular insurance, which entirely eliminates the element of risk. Evidently, the intuitive notion of risk is not adequately captured by the assumed concavity of the utility function for wealth The aversion for probabilistic insurance is particularly intriguing because all insurance is, in a sense, probabilistic. The most avid buyer of insurance remains ulnerable to many financial and other risks which his policies do not cover. There appears to be a significant difference between probabilistic insurance and what may be called contingent insurance, which provides the certainty of coverage for a