IA idea · Games & puzzles
What are the odds in a Risk battle?
Research question
What is the probability that an attacking army of size A defeats a defending army of size D in Risk, and how many attackers are needed for a 75% chance of success?
Adapt it: change the place, the data or the comparison until the question is yours.
Why it makes a good exploration
Each Risk roll compares the highest dice, which needs careful counting. Chaining rolls together into a Markov chain gives exact battle odds you can turn into a strategy.
The mathematics you'll need
- Probability distributions of the maximum of dice
- Counting outcomes (3 vs 2 dice: 7,776 cases)
- Markov chains or recursion for full battles
- Simulation to verify
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 official rules; play and record battles to compare.
- 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
- Compute single-roll outcome probabilities.
- Model a battle as a chain of rolls.
- Compute win probabilities for various armies.
- Find the attackers needed for 75%.
- Reflect on rule variants and strategy.
Pitfalls that cost marks
- Errors in counting ordered dice comparisons — check with a program.
- Ignoring that attackers can stop.
- Tables without interpretation.
Showing personal engagement
- Use your own games' data.
- Create a quick-reference table for players.
- Compare with other dice-battle games.
See Criterion C: personal engagement for what examiners look for.
Taking it further
Find the expected number of armies lost in a battle and use it to plan a campaign.