2017 VJC P2 Q10

It is a common practice for airlines to sell more plane tickets than the number of seats available. This is to maximise their profits as it is expected that some passengers will not turn up for the flight. The plane used by Victoria Airline for her daily 10 am flight from Singapore to Hong Kong has a maximum capacity of \(150\) seats. For this particular flight, \(154\) tickets are sold every day.

On average, \(p\) out of \(100\) customers who have purchased a plane ticket for this flight turn up. Customers who turn up after the flight is full will be turned away. The number of customers who turn up for the 10 am flight, on a randomly chosen day, is denoted by \(X\).

1. State, in the context of this question, two assumptions needed to model \(X\) by a binomial distribution.[2]
2. Explain why one of the assumptions stated in part (i) may not hold in this context.

Assume now that these assumptions do in fact hold.

3. It is known that there is a \(0.05\) probability that at least \(153\) customers will turn up for the 10 am flight. Write down an equation for the value of \(p\), and find this value numerically.[3]

It is given instead that \(p=94\).

4. Find the probability that, on a randomly chosen day,
   1. there are at least \(141\) but not more than \(148\) customers who turn up for the 10 am flight,[2]
   2. every customer who turns up gets a seat on the 10 am flight.[1]
5. Find the probability that every customer who turns up gets a seat on the 10 am flight on more than \(5\) days in a week.[3]