Maths IA · Statistics and probability · topic 17 of 40

Card and dice games: what is the best strategy? · IB Maths IA idea: guide, sample graph and step-by-step plan

gentle start, easy to deepen

Every game has a rule that decides whether the house or the player wins in the long run. You're going to find it and then test it by playing.

Difficulty●○○ Gentle
KitCards or dice, spreadsheet for simulation
TimeAbout 10 to 12 hours
FitsAA and AI, SL and HL
  • ✓ you like games and strategy
  • ✓ you enjoy tree diagrams
  • ✓ you want a gentle first project

The big idea

the long run beats a hunch

Pick a game with clear rules and some choices: stick or twist, which bet, when to stop. Use conditional probability to model each stage, then expected value to compare strategies. The answer might be that a popular strategy does not help, or that one rule change flips who has the edge. Back your model with a simulation or real plays, and decide how many plays are enough to trust the average. The fun comes from checking your theory against a long run of play. At HL, make the model deeper: several stages, a recursion, or a proof that a strategy is best.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Compare two strategies

Write down two ways to play, such as stop at 17 versus stop at 15, and compute the expected gain of each.

→ The analysis becomes a comparison of expected values, and your conclusion names the better one.
Change one rule

Alter the payout or add a house rule and see how the expected value moves.

→ You get a graph of expected gain against the changed parameter, with a break-even point.
Simulate a million plays

Code or use a spreadsheet to play thousands of rounds.

→ Compare simulated averages with exact expected values and discuss the law of large numbers.
Use a real game

A board game, a card game from home or a casino-style game with published rules.

→ Your introduction has a personal reason, and rules may be messier than a textbook example.
Long-run risk

Calculate the variance of the gain, or the chance of going broke from a starting budget.

→ Expected value is no longer enough; you must discuss spread and risk, which deepens the maths.
Optimal strategy for HL

Prove which choice at each stage is best, working backwards from the last stage.

→ This adds the rigour and sophistication HL needs; checking a few cases is not a proof.

Your Maths IA research question: run the check

Write your own question first. Then make sure it ticks every box:

  • Names one game with fully stated rules.
  • Says which choices a player has and which you will compare.
  • Says what you will measure: probability of winning, expected gain per play, or both.
  • Says how you will test your answer (simulation or real plays) and how many.
  • Can be answered with exact probabilities, not just 'play and see'.

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The picture to draw

draw the tree for YOUR game

A tree diagram for one round, with a payoff at the end of each branch. Yours will match your game's real rules.

4/5248/523/5148/514/5147/51payoff xx₁x₂x₃x₄A = an ace, N = not an aceE(X) = Σ x · P(X = x)does the game pay the fair price?no replacement:denominators change

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why this game, your own experience of it, what you think the best strategy is before calculating.
Rules and assumptions~1 page
The rules exactly as you will model them, plus what you assume (fair shuffle, fair dice, no counting cards).
Probability model~3 pages
Tree diagram or table with conditional probabilities, a worked example, and a check that the probabilities sum to 1.
Expected value~2 pages
Definition, worked calculation for each strategy, and a comparison in a table.
Simulation or real play~2 pages
How the simulation was built or how many real rounds were played, results table and graph against the exact value.
Going deeper~2 pages
Risk, an extra stage, a rule change or a proof of the best choice. At HL, this carries the sophistication.
Conclusion~1 page
Which strategy is best and by how much, with units (gain per play).
Reflection and limits~1.5 pages
Which assumption matters most, small-sample behaviour, whether real people play optimally, and one next step.
Bibliographynot counted
Rules source and any tools used.

What your Maths IA graph could look like

Made-up simulation averages, to show the form of a strong graph. example only

2345678910-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0Payout m for rolling a total of 7 (units of the stake)Average net gain per play (stake units)exact expected value, m/6 − 1break-evensimulated average of 200 plays (made up)where does the game become fair?What does a points cloud near theline tell you?ILLUSTRATIVE · MADE-UP DATA
  • Plot the exact value as a line and the simulated averages as points, so the reader sees the gap.
  • Error bars from the spread of the simulated averages (state how many plays each used).
  • A break-even line makes the key question visible.
  • Say clearly the payout rule being drawn; the units of the stake matter.
  • Another graph to consider: the running average of one long simulation against the number of plays.

Your game plan

play it by hand first

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Choose a game with choicesA game where players decide something: when to stop, which card to keep, which bet to place. Write out the rules precisely.
  2. PredictWrite down which strategy you think is best and whether the game favours the player, before any calculation.
  3. Learn the toolsConditional probability, tree diagrams, expected value. Do a one-round example of a simpler game by hand.
  4. Trial runPlay your game 30 times by hand, or run a quick spreadsheet. Do the rules produce situations you had not thought about?
  5. Build the exact modelTree diagram with conditional probabilities. Check each branch set sums to 1.
  6. Compute expected valuesFor each strategy, with a clear table. Say what the answer means in the game's own units.
  7. Simulate or playDecide how many plays are enough, and why. Compare the average with the exact expected value.
  8. Go one step deeperChoose risk, an extra stage, a rule change, or a proof. Justify the choice.
  9. ReflectWhat assumptions did you make? What would a real player do differently? Reflect throughout, not only at the end.
  10. Write, then cutOne worked tree, then tables. Keep notation consistent (X for payoff, not several letters) and check before your teacher's draft.

Maths you'll need

Traps to avoid in this Maths IA

  • A game so simple the probability is only a few lines; add a second stage or a comparison.
  • Forgetting that probabilities change without replacement.
  • Simulating only 20 plays and calling the average 'accurate'.
  • Treating expected value as what will happen in one play.
  • A strategy chosen without comparing it with at least one other.
  • At HL: a basic dice game with SL-level probability only; HL needs more depth, such as a proof, recursion or links to another topic.
  • Using a spreadsheet function name or computer notation in the written explanation.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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