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?

Adapt it: change the place, the data or the comparison until the question is yours.

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

Model the distribution of annual winnings for a person who plays every week.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).