This situation can be modeled as a 3-state Markov chain, where the states represent the 3 discount levels:
0% discount = State 1
30% discount = State 2
60% discount = State 3
The transitions between states depend on the number of claims in a year. Let's define the transition probability matrix as:
P = [p(i,j)]
where p(i,j) is the probability of moving from state i to state j.
We can write the transition probabilities as:
p(1,1) = e^-λ //probability of 0 claims, so stay in state 1
p(1,2) = 1 - e^-λ //probability of 1 claim, so move from state 1 to 2
p(2,1) = e^-λ //probability of 0 claims, so move from state 2 to 1
p(2,2) = 1 - e^-λ //probability of 1 claim, so stay in state 2
p(2,3) = 0 //cannot move from state 2 to 3 in 1 transition
p(3,2) = e^-λ //probability of 0 claims, so move from state 3 to 2
p(3,3) = 1 - e^-λ //probability of 1 claim, so stay in state 3
Therefore, the transition probability matrix is:
P = [[e^-λ, 1 - e^-λ, 0],
[e^-λ, 1 - e^-λ, 0],
[0, e^-λ, 1 - e^-λ]]