Under an insurance policy, a maximum of …five claims may be filed per year by a policyholder. Let $\displaystyle p_n$ be the probability that a policyholder files n claims during a given year, where n = 0,1,2,3,4,5. An actuary makes

the following observations:

(a) $\displaystyle p_n$ >= $\displaystyle p_{n+1}$ for n = 0, 1, 2, 3, 4.

(b) The difference between $\displaystyle p_n$ and $\displaystyle p_{n+1}$ is the same for n = 0, 1, 2, 3, 4.

(c) Exactly 40% of policyholders file fewer than two claims during a given year.

How would you calculate the probability that a random policyholder will …file more than three claims during a given year?