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?
Make it yours, then agree it with your teacher.
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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
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.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Probability distributions of the maximum of dice; Counting outcomes (3 vs 2 dice: 7,776 cases) |
| AA HL | Good fit | Probability distributions of the maximum of dice; Counting outcomes (3 vs 2 dice: 7,776 cases) |
| AI SL | Good fit | Probability distributions of the maximum of dice; Counting outcomes (3 vs 2 dice: 7,776 cases) |
| AI HL | Good fit | Probability distributions of the maximum of dice; Counting outcomes (3 vs 2 dice: 7,776 cases) |
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: errors in counting ordered dice comparisons — check with a program; and ignoring that attackers can stop. Say how each one affects your answer.
Use of mathematics (E)
SL: 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, 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 number of armies lost in a battle and use it to plan a campaign.
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?
Probability distributions of the maximum of dice, counting outcomes (3 vs 2 dice: 7,776 cases), markov chains or recursion for full battles and simulation to verify.
Where can I get data for this IA?
Use the official rules; play and record battles to compare. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
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