Problem source

At the terminus of a bus route, passengers arrive at an average rate
of 4 per minute according to a Poisson process. Each minute, on the
minute, one bus arrives with probability $\frac{1}{4},$ independently
of the arrival of passengers or previous buses. Just after eight o'clock
there is no-one at the bus stop. 
\begin{questionparts}
\item What is the probability that the first bus arrives at $n$ minutes
past 8?
\item If the first bus arrives at 8:05, what is the probability that there
are $m$ people waiting for it?
\item Each bus can take 25 people and, since it is the terminus, the bus
arrive empty. Explain carefully how you would calculate, to two significant
figures, the probability that when the first bus arrives it is unable
to pick up all the passengers. Your method should need the use of
a calculator and standard tables only. There is no need to carry out
the calculation. 
\end{questionparts}