Poisson Process with Conditional Probability

Question

It's told that buses arrive at a station according to a Poisson process with a rate of 5 per hour.
I am trying to find the probability that the fifth bus of the day arrives after 12 p.m. given the buses start arriving at 9 a.m.
I am stuck as I do not know how should I denote this and the condition in terms of probability: $$P((N(0,3)=4)| ? )$$ Can anyone possibly help me with this? Thank you!

Answer 1

Letting $N(t) = $ # of bus arrivals by hour $t$, and $S_i = i^{th}$ arrival time:

$? = P(24 \geq S_5 \geq 12 \ | \ N(9) = 0)$

Does that make sense?

Answer 2

Let $N(s,t)$ be the count for arrivals within interval $(s..t]$, with the time measured in hours.

Then we are given that $N(s,t)\sim\mathcal {Pois}(5(t-s))$.

The event that the fifth arrival occurs after 12pm, is the event that at most four arrivals have occurred prior than 12pm and some bus arrives there after.   That is $N(0,12)\leq 4$ and $N(12,24)>0$ .

The event that the arrivals start at 9am, is the event that no arrivals occur before that time, ie $N(0,9)=0$. [Though I am not sure if we should add that an arrival occurs exactly at nine.]

Thus you seek : $~\mathsf P(N(0,12)\leq 4, N(12,24)>0\mid N(0,9)=0)$

Use the definition of conditional probability to evaluate this.

[Note that the counts for arrivals within disjoint intervals are independent.]