Ferry overbooking and expected net revenue

Bluewater Ferries operates a service with \(180\) passenger seats. To reduce the number of empty seats, the company may accept more reservations than the number of seats available.

Each reservation costs \(\$72\) and is non-refundable. Past records show that a reservation holder does not attend with probability \(0.07\). If more than \(180\) reservation holders attend, the company pays \(\$320\) in compensation to each passenger who is unable to board.

Assume that reservation holders attend independently and that the attendance probability is the same for every reservation. For a particular crossing, the company accepts \(190\) reservations. Let \(X\) denote the number of reservation holders who attend.

1. Find the probability that there are insufficient seats for the crossing.[2]
2. Calculate the expected net revenue from the crossing, giving your answer to the nearest dollar.[3]

Suppose instead that the company accepts \(n\) reservations, where \(n>180\), and let \(X\sim\mathrm{B}(n,0.93)\).

3. Write down an expression, in terms of \(n\) and binomial probabilities, for the expected net revenue from one crossing.[2]
4. Hence, determine the number of reservations that the company should accept to maximise the expected net revenue, and state this maximum expected net revenue to the nearest dollar.[3]