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!