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.