CAS Exam 1Probability (P)

30% of drivers are high-risk with an annual accident probability of 0.20; the remaining 70% are low-risk with probability 0.05. A randomly chosen driver has an accident this year. Calculate the probability the driver is high-risk.

Let $X_1,\dots,X_n$ be i.i.d. Uniform$(0,\theta)$. Derive the distribution of $M=\max_i X_i$ and its expected value. Comment on whether $M$ is an unbiased estimator of $\theta$ and propose a correction.

Auto policyholders are classified as Preferred (50% of book, annual claim probability 0.02), Standard (30%, probability 0.05), or Substandard (20%, probability 0.10). A policyholder has a claim this year. Calculate the probability the policyholder is Substandard.