I think I'm getting confused when looking at conditional probability questions.
I have just looked at a question where there are 2 groups of policy holders. The probability of being in group A is 60% and group B 40%. The number of claims submitted for A follows a Poisson Distribution with lambda = 0.2. The number of claims submitted for A follows a Poisson Distribution with lambda = 0.5. It states that an individual policyholder submits claims independently of claims submitted by other policyholders. The number of claims submitted in a given year is independent of claims submitted in previous years.
The question asked that given we don't know which group a policyholder belongs to, determine the probability that a randomly chosen policyholder will submit one claim this year, given that they submitted one claim last year.
When I tried to answer this, I thought that since the claims were independent across years that: P(X=1 this year|X=1 last year) = P(X=1 this year).
I spent a while trying to find an answer online and got something along the lines of 'since we don't know which group the policy holder belongs to, we cannot say the claims are independent across years'. I still don't really understand this and was hoping that someone might be able to explain a little further!
Also if we did know the group that the policyholder belonged to, then would that answer be correct?
Sorry for the long post - any help is very much appreciated!
Thanks!