A service station handles jobs of two types, A and B. (Multiple jobs can be processed simultaneously.) Arrivals of the two job types are independent Poisson processes with parameters $λ_A = 3$ and $λ_B = 4$ per minute, respectively. Type A jobs stay in the service station for exactly one minute. Each type B job stays in the service station for a random but integer amount of time which is geometrically distributed, with mean equal to 2, and independent of everything else. The service station started operating at some time in the remote past.
Question: At time $0$, no job is present in the service station. What is the PMF of the number of type B jobs that arrive in the future, but before the first type A arrival?
I was able to correctly arrive at the answer $p_K(k) = (\frac{3}{7}) (\frac{4}{7})^k, k = 0,1,2...$ by reasoning that there was a $4/7$ probability that an arrival was from $B$ and if we have $k$ type $B$ jobs before the first type $A$ then that gives a probability of $(\frac{3}{7}) (\frac{4}{7})^k$. However, other than seeing the official answer agrees with mine, I am not too convinced of my answer. Is there a better way to reason to the answer (with no measure theory)?