Problems

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.

1. What is the probability that the first bus arrives at \(n\) minutes past 8?
2. If the first bus arrives at 8:05, what is the probability that there are \(m\) people waiting for it?
3. 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.