1. **Problem statement:** We want to find the probability that an insured who did not claim last year will not claim in the next 5 years. 2. **Define states:** Let: - $N$: no accident occurred in the last year. - $A$: accident occurred in the last year. 3. **Transition probabilities given:** - $P(N \to A) = 0.04$ (4% chance of accident this year if no accident last year) - $P(N \to N) = 1 - 0.04 = 0.96$ - $P(A \to A) = 0.12$ (12% chance of accident this year if accident last year) - $P(A \to N) = 1 - 0.12 = 0.88$ 4. **Initial state:** The insured did not claim last year, so we start in state $N$. 5. **Goal:** Find $P( ext{no claim in next 5 years} \mid ext{start in } N)$. "No claim" means staying in state $N$ every year for 5 years. 6. **Calculate step by step:** We want the probability that each year has no accident given the state transitions: - Start at $N$ with probability 1. - Year 1 no claim: $P(N \to N) = 0.96$ - Year 2 no claim: depends on previous year's state: - Last year was $N$, next year no claim: $0.96$ - Last year was $A$, next year no claim: $0.88$ But to have no claim in both years, we must consider the joint probabilities of paths with no accidents. 7. **Derive probability of no claims recursively over 5 years:** Let: - $p_n(N)$ = Probability of no claim in year $n$ and being in state $N$ at year $n$. - $p_n(A)$ = Probability of no claim in year $n$ and being in state $A$ at year $n$. However, since no claim means staying in $N$, $p_n(A) = 0$ for all $n$. So from state $N$: - $p_1 = P(N \to N) = 0.96$ - $p_2 = p_1 \times 0.96 = 0.96^2$ - $p_3 = 0.96^3$ - $p_4 = 0.96^4$ - $p_5 = 0.96^5$ The result is because any accident ($A$) means a claim, so to have no claims, the insured must remain in $N$ all years. 8. **Final answer:** $$ P(\text{no claim in 5 years} | N) = 0.96^5 = (0.96)^5 $$ Calculating numerically: $$ 0.96^5 = 0.8153726976 \approx 0.8154 $$ So the probability is approximately 0.8154 or 81.54%.