A busy railway station has two taxi stands at separate exits, A and B. At stand A, taxis arrive according to a Poisson Process of rate 2 per minute.
Assume that any arriving train passenger who requires a taxi goes to stand A with probability 0.7 or to stand B with probability 0.3. At stand B, taxis arrive according to a Poisson Process of rate 1 per minute.
Three minutes after reaching the front of the queue at one of the taxi stands, a train passenger finds herself still waiting for a taxi. Given this information, find the probability that this passenger is waiting at taxi stand A.
I've spent some time thinking about this question, though nothing much really comes to mind.
Part of me believes I should use all the information given, however I'm having a hard time linking the given information to what we need to find.
Some suggestions would be helpful...
Guide:
Let $T$ be the event that you wait for more than $3$ minutes \begin{align} P(A|T) = \frac{p(A)P(T|A)}{P(T)} \end{align}