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

Is a lottery ticket ever worth buying? Expected value and rollovers

AA SLAA HLAI SLAI HL Accessible Also in: Finance

Research question

For [a national lottery], what is the expected value of a ticket, and how large must a rollover jackpot be for the expected value to exceed the ticket price once shared jackpots are considered?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Lotteries are designed to have negative expected value — but rollovers change the numbers. Working out when a ticket is “fair”, and why it still is not, uses combinations, expected value and a clever Poisson approximation for shared wins.

The mathematics you'll need

Combinations for the probability of each prize
Expected value and variance
Poisson approximation for the number of other jackpot winners (AI HL)
Break-even analysis

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Binomial and Poisson models. 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 the lottery's published rules, prize structure and odds, and published ticket sales for rollover draws.

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

A possible outline

Calculate the probability of each prize tier from first principles.
Compute the expected value of a normal draw.
Include rollovers and shared jackpots.
Find the break-even jackpot.
Reflect on variance, utility and tax.

Pitfalls that cost marks

Copying odds without deriving them.
Ignoring shared jackpots.
Treating expected value as the only thing that matters.

Showing personal engagement

Compare two lotteries with different rules.
Discuss why people play anyway (utility of a tiny chance).
Survey adults about how much they spend (anonymously).

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	Combinations for the probability of each prize; Expected value and variance
AA HL	Fits, but add an HL technique	Combinations for the probability of each prize; Expected value and variance
AI SL	Good fit	Combinations for the probability of each prize; Expected value and variance
AI HL	Fits, but add an HL technique	Combinations for the probability of each prize; Expected value and variance

Level: Accessible. A good first extended piece of maths, with room to go deeper. 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: copying odds without deriving them; and ignoring shared jackpots. Say how each one affects your answer.

Use of mathematics (E)

SL: combinations for the probability of each prize; expected value and variance; break-even analysis, each explained in your own words and carried out correctly.

HL adds: poisson approximation for the number of other jackpot winners, 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 the distribution of annual winnings for a person who plays every week.

Extending it for HL

This idea already has HL mathematics in it: Poisson approximation for the number of other jackpot winners. Make that the centre of the exploration and use it with full rigour.

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 Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Combinations for the probability of each prize, expected value and variance, poisson approximation for the number of other jackpot winners (AI HL) and break-even analysis.

Where can I get data for this IA?

Use the lottery's published rules, prize structure and odds, and published ticket sales for rollover draws. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Is a lottery ticket ever worth buying? Expected value and rollovers” filled in: the question, the maths and the data sources.

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