IB Mathematics Internal Assessment
Cards, dice, board games, poker, video game loot drops. Games are easy to care about and easy to explain, and probability gives you real mathematics to do. They also produce some of the thinnest explorations I read, because a game rule can be turned into a number in one line. Here is how to give the mathematics a proper job.
You already know the rules, so there is no setting-up cost, and personal engagement feels automatic. The trap is that the answer to most game questions can be found with a single formula or a quick simulation, and then the exploration has nowhere left to go.
The question to keep asking is this: what would I not have known if I had not done the mathematics? If your conclusion is something every player already believes, such as "the house always wins", the work has to show how and why, and by how much, not just that.
List the possible hands or rolls, divide favourable by total, and report a probability. This is routine counting, and at HL it sits well below the expected level. There is no decision in it and nothing that could surprise you.
A short program does all the work, and you describe the output. If you cannot say why the simulation settles down, how many runs you needed and what the spread of results looks like, the program has done the mathematics and you have done the screenshots.
Popular, and usually a foregone conclusion. If every bet has the same expected loss, no staking plan changes that, and a page of tables repeating it adds little. It can work only if you prove or test something specific about how the plan behaves, rather than announcing the ending in the title.
The prisoner's dilemma, rock paper scissors, a payoff table lifted from a website. Reproducing a standard result shows you can read, not that you can explore. The mathematics has to be applied to a situation of your own.
Choose a game where a player faces real choices, such as when to stop, which option to take or how much to risk. Define what "better" means, calculate the expected outcome of each strategy, then ask whether expected value is even the right measure for someone who plays only once.
Model the squares as states and the dice as transition probabilities, then find which squares are visited most and how long a game lasts. You build the matrix yourself, check it against real play, and work out why it differs. This suits AA students, and AI students comfortable with matrices.
In a game with hidden information, such as cards held by others, how should your estimate change as play reveals more? Set out the conditional structure carefully, calculate by hand for a small case, and test your reasoning against recorded hands. Keep the case small enough to check honestly.
Collect your own data from real dice, a shuffled deck or a spinner, and test whether it matches the model. A chi-squared test can belong here, but only if you can explain why it suits your data. Be ready for an inconclusive result, and say what that means for your claim.
Choose the rules so a game lasts about a given time, or so each player has a stated chance of winning. You set the aim, vary the parameters, solve for the values that work and then play-test the result. The design decisions are yours, which makes the engagement easy to evidence.
Take "the best strategy for a dice game where you can keep rolling or bank your score" as a starting idea. Calculated once, it is a short exercise. To turn it into an exploration, keep pushing.
Nothing there needs advanced mathematics. It needs a question you can defend and an honest look at your own result.
Reflection comes naturally with games, because real play will often disagree with your model. Record those moments properly, with what you expected, what you found and what you changed. They are worth more than another table of probabilities.
Got a game idea and not sure it can score?
Send me the question and what you plan to do with the mathematics. I read real submitted explorations every year, and I will tell you honestly whether it has enough to do, while there is still time to change direction.
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