2022 HSC Mathematics Extension 1
Question 14(d):
Binomial distribution
An airline finds a probability that a booked passenger misses the flight. Management allows overbooking but requires that no more than of flights have more passengers showing up than seats.
Using a suitable approximation, find the maximum number of tickets that can be sold for a flight with 350 seats.
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The requested response is a worked solution.
How to handle it: identify the random variable and its parameters before using the matching probability formula.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Let be tickets sold and the number who show up. Require , i.e. .
- Approximate by with , .
- Without continuity correction, need .
- Solve . Set or treat as quadratic in : .
- NESA sample yields , so the largest integer ticket count meeting the risk constraint is (checking exceeds the risk).
- With continuity correction the same integer is obtained ( then floor/check).
- Answer: 358 tickets.
- NESA: 4 marks; 3 for a quadratic inequality in ; 2 for a normal approximation setup; 1 for a probability statement like .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Seats 100, show-up probability 0.9, allow at most 5% chance of overflow. Outline the normal approximation inequality for tickets .
Question 2
If and , find and the number of standard deviations from 350.
Question 3
Translate 'no more than 1% of flights overfull' into a statement about for a 350-seat flight.