The answer is 0.223584.
Here is my attempt:
Company A: $\mu = 10000\\ \sigma = 2000\\ \text{40% chance of at least one claim}$
Company B: $\mu = 9000\\ \sigma = 2000\\ \text{30% chance of at least one claim}$
Then $P[A < B] = P[\frac{A - 10000}{2000} < \frac{B - 9000}{2000}] = P[A - B < 1000]$.
From here, I thought it would be finding the area of the region described by $A - B < 1000$, but I wasn't sure if that was correct. Also, I am not sure how to incorporate the fact that there is a 40% and 30% chance of companies A and B filing at least one claim.
Let $B$ be the total claim amount for company B and $A$ the total claim amount for company A. To calculate $P(B>A)$, you condition on whether $A>0$ and $B>0$: $$ \begin{align} P(B>A)&=P(B>A|A=0,B=0)P(A=0,B=0)\\ &+P(B>A|A>0,B=0)P(A>0,B=0)\\ &+P(B>A|A=0,B>0)P(A=0,B>0)\\ &+P(B>A|A>0,B>0)P(A>0,B>0)\\ &= (1) + (2) + (3) + (4) \end{align} $$ Handle each of the terms (1), (2), (3), (4) separately. The first 2 terms are zero (why?). The third simplifies to $ 1 \times P(A=0)P(B>0)$. The fourth term requires you to calculate $P(X>0)$, where $X$ is the distribution of $B-A$ conditional on both $B$ and $A$ being positive (so $X$ is normally distributed with mean = $\mu_B-\mu_A = -1000$ and variance = $\sigma^2_B + \sigma^2_A = 2000^2 + 2000^2$), and then multiply that probability by $P(A>0)P(B>0)$.