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

What are the chances of rolling a Yahtzee in one turn?

AA SLAA HLAI SLAI HL Solid Also in: Probability, Networks & graphs

Research question

With the best re-roll strategy, what is the probability of getting five of a kind within three rolls, and how does it compare with simulation and my own games?

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Why it makes a good exploration

The answer (about 4.6%) is not obvious because you keep dice between rolls. Modelling the turn as a chain of states (number of matching dice) is a neat, self-contained probability problem.

The mathematics you'll need

Binomial probabilities
States and transition probabilities (a Markov chain)
Matrix multiplication or tree diagrams
Simulation and experimental verification

Where the data comes from

Play 100+ turns aiming for five of a kind and record results; simulate many more.

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

A possible outline

Define the states and strategy.
Compute transition probabilities.
Combine three rolls to get the answer.
Compare with simulation and play.
Reflect on strategy choices and extensions.

Pitfalls that cost marks

Ignoring that the most common value may change between rolls.
Too few experimental turns.
Unclear strategy.

Showing personal engagement

Play with family and record the results.
Compute probabilities for other scoring categories.
Decide when to give up on Yahtzee using expected points.

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 probabilities; States and transition probabilities (a Markov chain)
AA HL	Good fit	Binomial probabilities; States and transition probabilities (a Markov chain)
AI SL	Good fit	Binomial probabilities; States and transition probabilities (a Markov chain)
AI HL	Good fit	Binomial probabilities; States and transition probabilities (a Markov chain)

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: ignoring that the most common value may change between rolls; and too few experimental turns. Say how each one affects your answer.

Use of mathematics (E)

SL: binomial probabilities; states and transition probabilities (a Markov chain); matrix multiplication or tree diagrams; simulation and experimental verification, each explained in your own words and carried out correctly.

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

Taking it further

Find the expected score of a turn aimed at a large straight and compare strategies.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL games & puzzles 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 games & puzzles 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 probabilities, states and transition probabilities (a Markov chain), matrix multiplication or tree diagrams and simulation and experimental verification.

Where can I get data for this IA?

Play 100+ turns aiming for five of a kind and record results; simulate many more. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “What are the chances of rolling a Yahtzee in one turn?” filled in: the question, the maths and the data sources.

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