Question

Let pn, n = 0,1,2,..., be the probability that an automobile policyholder will file for n claims in a five-year period. the actuary involved makes the assump- tion that pn+1 = (1/4)pn. what is the probability that the holder will file two or more claims during this period?

Answer 1

P(1 claim) = p/4
P(2 claims) = (p/4)/4 = p/16

You should see that the distribution follows a geometric series with common ratio 1/4.
Sum geometric = (first term) / (1 - common ratio) = p/(1 - 1/4) = 4p/3

But the sum of all the probabilites must equal 1 ----> 4p/3 = 1 ----> p = 3/4

P(2 or more claims) = 1 - P(0 claims) - P(1 claim) = 1 - 3/4 - 3/16 = 1/16