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

Radioactive decay with dice: does chance produce an exponential curve?

AA SLAI SLAA HLAI HL Accessible Also in: Modelling, Differential equations

Research question

When 200 dice are rolled repeatedly and every six is removed, does the number remaining follow N = N₀(5/6)ⁿ, and how closely does the experimental half-life match the theoretical one?

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Why it makes a good exploration

No atom “knows” when to decay, yet a large sample decays exponentially. Rolling dice reproduces this exactly — and the randomness in your results is itself something to analyse.

The mathematics you'll need

Exponential models and half-life
Binomial distribution of the number removed each round
Logarithms to find the experimental decay constant
Expected value and variance; spread between trials

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

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

Roll 200 dice (or simulate with a spreadsheet) and remove sixes; repeat the whole experiment several times.

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

A possible outline

Derive the expected model from p = 1/6.
Collect several decay runs.
Fit an exponential; compare the decay constant with ln(6/5).
Examine run-to-run variation using the binomial distribution.
Link to carbon dating and real half-lives.

Pitfalls that cost marks

One run only (you cannot discuss randomness).
Confusing the discrete and continuous models.
Fitting without explaining the log transformation.

Showing personal engagement

Try dice with different faces (d4, d20) and predict each half-life.
Compare your class's pooled data with individual runs.
Explain why small samples decay erratically.

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	Exponential models and half-life; Binomial distribution of the number removed each round
AA HL	Fits, but add an HL technique	Exponential models and half-life; Binomial distribution of the number removed each round
AI SL	Good fit	Exponential models and half-life; Binomial distribution of the number removed each round
AI HL	Fits, but add an HL technique	Exponential models and half-life; Binomial distribution of the number removed each round

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: one run only (you cannot discuss randomness); and confusing the discrete and continuous models. Say how each one affects your answer.

Use of mathematics (E)

SL: exponential models and half-life; binomial distribution of the number removed each round; logarithms to find the experimental decay constant; expected value and variance; spread between trials, 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

Use your model to explain the uncertainty in radiocarbon dating of a small sample.

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, AI SL, AA HL 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?

Exponential models and half-life, binomial distribution of the number removed each round, logarithms to find the experimental decay constant and expected value and variance; spread between trials.

Where can I get data for this IA?

Roll 200 dice (or simulate with a spreadsheet) and remove sixes; repeat the whole experiment several times. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Radioactive decay with dice: does chance produce an exponential curve?” filled in: the question, the maths and the data sources.

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