Question Regarding Poisson and probability. i found this interesting question on the web but i am not quite sure if my solution is accurate. Honestly i would appreciate few opinions.
Given Question:

At a subway station, eastbound trains and northbound trains arrive
  independently, both according to a Poisson process. On average, there
  is one eastbound train every 12 minutes and one northbound train
  every 8 minutes.
  Suppose you arrive at the subway station at a certain point in time and
  start observing trains.
  Question: Find the probability at most  2 trains to reach the station within the next 10 minutes. 

So my attempted solution is pretty straight forward: I have calculated the Probability for EastBound train and then the probability for NorthBound train. 
So, where i am really stuck here is whether should i add those two probabilities or multiply them! For example i have calculated the probability for eastbound train as follows (in a very draft manner, please excuse me)  $$\Bbb{P}_{east}= \big\{\Bbb{P}(N(10)=0)+\Bbb{P}(N(10)=1)+\Bbb{P}(N(10)=2)\big\}$$    .Thanks for the imput!
 A: We want the sum of the following probabilities:
East $0$, North $0$ 
East $0$, North $1$
East $1$, North $0$
East $1$, North $1$
East $0$, North $2$
East $2$, North $0$.
For each of the lines, multiply the two relevant probabilities. Then add up. You will need the correct parameters for the two Poisson.  
But there is a nicer way. It turns out that the sum of two independent 
Poisson random variables with parameters $a$, $b$ is a Poisson with parameter $a+b$. 
Do it both ways, and the result mentioned in the preceding paragraph will not be mysterious anymore.
A: About our sum-or-multiply doubt: the basic rule is: if $A$ and $B$ are two mutually exclusive events (they cannot occur both simultaneously), then  the probability that $A$ or $B$ ocurrs is the sum: $P(A \cup B)=P(A)+P(B)$
So, your equation is right if the event you are computing is the "at most two eastbound trains reach the station in the next 10 minutes". But this might not be very useful to answer the question. Because you need to consider also the other train. See André's answer.
