In the realm of decision theory, the Sleeping Beauty Problem has long intrigued philosophers and scientists alike. This thought experiment poses a puzzling scenario where a subject is put to sleep and wakes up to find herself participating in a fair coin toss, with her memory erased. The crux of the problem lies in the interpretation of probability and how it affects decision-making. Similarly, in the world of insurance, the relationship between insurance excess and premiums can confuse the intuition of the insured.

The Sleeping Beauty Problem poses a scenario where a beauty goes to sleep and flips a fair coin. If it lands heads, she is awakened on Monday and interviewed. If it lands tail, she is awakened on Monday and Tuesday, interviewed both times and then the experiment ends. Regardless of the outcome, the beauty is asked, "What is your subjective probability that the coin landed heads?" This question presents a dilemma as the beauty struggles to assign a subjective probability due to her uncertainty about whether it is Monday or Tuesday. The conflicting viewpoints of different observers regarding the probability distribution for this problem have sparked debates among philosophers and statisticians. The two opposing viewpoints are the "Thirder" and the "Halfer". The Thirder argues that the probability is 1/3, while the Halfer believes it is 1/2. The controversy stems from the subjective interpretation of the probability of an event with limited information.

Thirder Position: The thirder position states that the probability of heads is 1/3. This is based on the principle of indifference, where if Sleeping Beauty believes the coin landed tails, her credence for Monday and Tuesday should be equal. Therefore, P (Tails and Tuesday) = P (Tails and Monday) = P (Heads and Monday), leading to all three outcomes having a probability of 1/3.

Halfer Position: The halfer position argues that Sleeping Beauty's credence for heads should be 1/2. Since she receives no new information during the experiment, her initial credence of P(Heads) = 1/2 should remain the same. This contradicts the thirded’s premise.


Insurance Excess and Premiums:

Now let's delve into the insurance realm and examine the intriguing relationship between insurance excess and premiums. Insurance policies often require policyholders to pay a certain amount, known as the excess or deductible, before the insurer covers the remaining costs. The purpose of the excess is to encourage responsible behavior and discourage the filing of small claims. However, when calculating premiums, a situation arises in which policyholders often do not understand that there is no linear relationship between the deductible and the premium. Small claims, such as minor accidents or repairs, are more frequent compared to large claims. The rationale behind this phenomenon lies in the statistical probability of occurrences. Minor incidents happen more frequently due to everyday mishaps, while major incidents are relatively rare events. Consequently, policyholders are more likely to encounter situations that require them to pay the excess for small claims rather than large claims. Given the higher frequency of small claims, it might seem intuitive that if the insurance excess is divided, the premium would increase proportionally and the other way around.

When the excess is changed, it primarily affects small claims, as large claims already exceed the initial excess amount. Therefore, the change impact more in excess payments for small claims, leading to a non-proportional reduction in premiums. Insurance companies consider the overall risk exposure and adjust premiums accordingly, determining the likelihood of various claim sizes.


The Sleeping Beauty Problem and the relationship between insurance excess and premiums share intriguing parallels. In both cases, the subjective interpretation of probability and the consideration of different scenarios play a significant role. While the Sleeping Beauty Problem challenges our understanding of probability, the non-proportional reduction in premiums resulting from changing the insurance excess highlights the complex nature of risk assessment in insurance. Understanding these concepts can help insurance companies to better explain to the insured the logic of the rates and coverage.