Let Pn=0,1,2....be the probability that an automobile policyholder will file for n claims in a five-year period. The Actuary involved make the assumption that Pn+1=(1/4)Pn. What is the probability that the holder will file two or more claims during this period?
Ah, probabilities and claims, always a fun combination! So let's calculate the probability that the policyholder will file two or more claims during this period, shall we?
Given the assumption that Pn+1 = (1/4)Pn, we can use this recursive equation to find the probabilities for each claim count. Starting with P0 = 1 (since the policyholder must file at least 0 claims), we can calculate P1, P2, P3, and so on.
P1 = (1/4)P0 = 1/4
P2 = (1/4)P1 = (1/4)(1/4) = 1/16
P3 = (1/4)P2 = (1/4)(1/16) = 1/64
Now, we need to find the sum of probabilities for two or more claims, which is P2 + P3 + P4 + ...
To find this sum, we notice that it is a geometric series with the first term P2 = 1/16 and the common ratio r = P3/P2 = (1/64)/(1/16) = 1/4. Using the formula for the sum of an infinite geometric series, the probability of two or more claims can be calculated as:
P(2 or more) = P2 / (1 - r) = (1/16) / (1 - 1/4) = (1/16) / (3/4) = 1/12
Therefore, the probability that the policyholder will file two or more claims during this five-year period is approximately 1/12.
Remember, this is just a mathematical assumption, so don't go scaring your policyholders with clownish probabilities!
To find the probability that the policyholder will file two or more claims during the five-year period, we can use the given recursive relation, Pn+1 = (1/4)Pn, along with the initial condition P0 = 1.
Since Pn represents the probability of filing n claims, we can rewrite the recursive relation as P1 = (1/4)P0, P2 = (1/4)P1, P3 = (1/4)P2, and so on.
Using these relations, we can calculate the probabilities for P1, P2, P3, and so on until we reach the desired probability.
P1 = (1/4)P0 = (1/4)(1) = 1/4
P2 = (1/4)P1 = (1/4)(1/4) = 1/16
P3 = (1/4)P2 = (1/4)(1/16) = 1/64
Continuing this pattern, we can observe that for any n > 1, Pn = (1/4)Pn-1.
Now, to find the probability of filing two or more claims, we need to sum up the probabilities from P2 onwards.
Probability = P2 + P3 + P4 + ...
To calculate this infinite sum, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r),
where "a" is the first term and "r" is the common ratio.
In this case, the first term is P2 = 1/16 and the common ratio is r = (1/4).
Sum = (1/16) / (1 - 1/4)
= (1/16) / (3/4)
= (1/16) * (4/3)
= 1/12
Therefore, the probability that the policyholder will file two or more claims during the five-year period is 1/12.