Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How many extra tickets should an airline sell?

AA SLAA HLAI SLAI HL Solid Also in: Finance, Statistics

Research question

For a 180-seat flight with a given no-show rate, how many tickets maximise the airline's expected revenue once compensation for bumped passengers is included?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Airlines deliberately sell more seats than they have. The binomial distribution and expected value give a clear optimum — and the ethics of bumping passengers gives you something real to reflect on.

The mathematics you'll need

Binomial distribution of passengers who show up
Expected revenue as a function of tickets sold
Maximising a discrete function
Normal approximation (optional)

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Binomial and Poisson models · Is my data normal?. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Use a published no-show rate range and published compensation rules (e.g., EU Regulation 261/2004 amounts) with citations.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Set up the revenue and compensation model.
Compute expected profit for each number of tickets sold.
Find the optimum and test sensitivity to the no-show rate.
Compare risk (probability of bumping anyone).
Reflect on independence (families and groups travel together).

Pitfalls that cost marks

Treating passengers as independent without comment.
Using a no-show rate with no source.
Ignoring the reputational cost.

Showing personal engagement

Use a route you have flown.
Include groups travelling together and see how the answer changes.
Discuss fairness from the passenger's point of view.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Binomial distribution of passengers who show up; Expected revenue as a function of tickets sold
AA HL	Good fit	Binomial distribution of passengers who show up; Expected revenue as a function of tickets sold
AI SL	Good fit	Binomial distribution of passengers who show up; Expected revenue as a function of tickets sold
AI HL	Good fit	Binomial distribution of passengers who show up; Expected revenue as a function of tickets sold

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: treating passengers as independent without comment; and using a no-show rate with no source. Say how each one affects your answer.

Use of mathematics (E)

SL: binomial distribution of passengers who show up; expected revenue as a function of tickets sold; maximising a discrete function; normal approximation (optional), each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Model group bookings with a compound distribution via simulation.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL probability exploration adds.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete probability exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Binomial distribution of passengers who show up, expected revenue as a function of tickets sold, maximising a discrete function and normal approximation (optional).

Where can I get data for this IA?

Use a published no-show rate range and published compensation rules (e.g., EU Regulation 261/2004 amounts) with citations. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How many extra tickets should an airline sell?” filled in: the question, the maths and the data sources.

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