I have three jobs $A,B,C$ with average arrival rates $a,b,c$. Each is independent poisson process. System waits until $10$ jobs arrive and then buffers them. Determine the probability that exactly two of each of the three types of jobs arrive during any 5 minute time interval.
Suppose that this interval is discretized into $20$ seconds each. Find the probability that two of each of three types of jobs can arrive in each interval. {Assuming at most one job can arrive in each interval}.
First part of question is quite easy.
$P(\textrm{"2 As arrive in 5 min interval"}) \cdot P(\textrm{"2 As arrive in 5 min interval"})\cdot P(\textrm{"2 As arrive in 5 min interval"})$.
Each one is poisson so just plugging in values into poisson's formula.
$$P(\textrm{"2 As arrive in 5 min interval"}) =\frac{(5a)^2 e^{-5a}}{2}$$ and similarly for the rest.
However, if 5 min interval is discretized into $20$ seconds then I have $15$ intervals. Now, this is a bernoulli process. How can I use Binomial to show that out of $15$ $2$ will have job $A$, $2$ will have job $B$ and $2$ will have job C.
For the reference, see question 1 last part: Problem Set