IA idea · Games & puzzles
What are the chances of rolling a Yahtzee in one turn?
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?
Adapt it: change the place, the data or the comparison until the question is yours.
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
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
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.
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
- 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.
Taking it further
Find the expected score of a turn aimed at a large straight and compare strategies.