## What’s the probability that exactly $12$ buses will arrive within $12$ hours

Let’s suppose there are two buses number $$86$$ and $$98$$. They draw up at the bus stop under the Poisson distribution with intensities $$3$$ and $$5$$ times per hour. (a) What’s the expected length of time after the $$15$$th bus will arrive?, (b)What’s the probability that exactly $$12$$ buses will arrive within $$12$$ hours?

Poisson distribution $$P(N(t)=j)=\frac{(\lambda t)^j}{j!}e^{-\lambda t}$$. We have that $$j =3$$ or $$j =5$$. Do I just substitute $$j =3$$ and $$\lambda t=15$$ and we immediately have (a)? I’m aware that’s a really easy exercise but I somehow don’t really know how to approach this one. I’m also not sure how to approach subpoint (b). I’ll be thankful for any tips and help.

## Do buses run on May 1st in Bavaria?

I am interested in landing in Munchen on 1st of May, and then take the bus to Ingolstadt. Does the bus operate in 1 May (since I am only interested in Airport-ZOB Ingolstadt)?

I guess it will be sparser, but that’s OK. I don’t want to book a flight and then find out that there is no bus! I pretended booking a ticket for the bus, but the system says its far too early, so doesn’t really answer the question.