IA ideas library
Probability IA ideas: chance, risk and expected value
Probability explorations can be done with no data collection at all — the mathematics is the investigation. You set up a model of a random process, calculate probabilities or expected values exactly, then check the model with a simulation or real results.
The strongest probability IAs ask a question someone would actually care about: is this game fair, how many attempts will it take, is this strategy better than that one?
Next step: shortlist ideas with the free IA topic explorer, which filters starting points by course, interest and how easy the data is to get. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.
How probability explorations reach the top bands
Personal engagement (C)
Play, simulate or collect the game or situation yourself. Predicting the answer before calculating, then checking it with an experiment, is engagement an examiner can see.
Reflection (D)
Compare theory with simulation or real data and explain differences. Question the assumptions of independence and equal likelihood.
Use of mathematics (E)
SL: A probability model built step by step (tree diagrams, binomial or normal distributions, expected value), used correctly, with the result checked by simulation or data.
HL: HL probability such as Bayes' theorem, Poisson processes, Markov chains, continuous random variables or a general result proved for n, not just computed for one case.
An HL version
Generalise the result to n (players, doors, dice) and prove it, or model the process as a Markov chain or with a continuous distribution.
Probability & chance ideas (13)
- How much does a small edge on each point matter in tennis?AA SLAA HLAI SLAI HLSolid
- Is a penalty shoot-out fair to the team that shoots second?AA SLAA HLAI SLAI HLSolid
- Radioactive decay with dice: does chance produce an exponential curve?AA SLAI SLAA HLAI HLAccessible
- The birthday problem, extended: what about birthdays within a day?AA SLAA HLAI SLAI HLSolid
- How many packs does it take to finish a sticker album?AA SLAA HLAI SLAI HLSolid
- Are arrivals at the school canteen random? A Poisson model of a queueAI HLAI SLSolid
- When should you stop looking? The 37% rule for choosing the bestAA HLAA SLAI HLAmbitious
- How many extra tickets should an airline sell?AA SLAA HLAI SLAI HLSolid
- Is the “hot hand” real? Streaks in my free throwsAI SLAI HLAA SLAA HLSolid
- Estimating π by dropping needles: how fast does the estimate improve?AA HLAA SLAI SLAI HLSolid
- The Monty Hall problem when the host is forgetfulAA SLAA HLAI SLAI HLSolid
- Is a lottery ticket ever worth buying? Expected value and rolloversAA SLAA HLAI SLAI HLAccessible
- What is the chance of hitting every green light on my route?AI SLAI HLAA SLAA HLAccessible
Also relevant (42)
- Is home advantage in football disappearing?AI SLAI HLAA SLAA HLAccessible
- Do left-handed tennis players have an advantage?AI SLAI HLAA HLSolid
- Does Benford's law hold for the populations of the world's towns?AI SLAI HLAA HLSolid
- Are elite footballers more likely to be born early in the selection year?AI SLAI HLAA SLAA HLAccessible
- Is the euro–dollar exchange rate a random walk?AI SLAI HLAA HLAmbitious
- Are cricket batting scores geometrically distributed?AI SLAI HLAA HLAA SLSolid
- Can a simple rating system predict football results?AI SLAI HLAA HLAmbitious
- Where should I aim on a dartboard?AI SLAI HLAA HLAmbitious
- Are Atlantic hurricanes becoming more frequent?AI SLAI HLAmbitious
- A positive test result: how likely is it that you really have the condition?AA HLAA SLAI SLAI HLSolid
- How does Google rank pages? PageRank on a small websiteAI HLAA HLAmbitious
- Will one phone brand take over? Brand loyalty as a Markov chainAI HLSolid
- The Collatz conjecture: what can we say about stopping times?AA SLAA HLAI SLAI HLSolid
- How long does a game of Snakes and Ladders last?AI HLAA HLAmbitious
- Which Monopoly squares are landed on most often?AI HLAA HLAmbitious
- How big is White's advantage in chess, and does it depend on skill?AI SLAI HLAA HLSolid
- Can you beat people at rock–paper–scissors?AI SLAI HLAA SLAA HLAccessible
- What are the chances of rolling a Yahtzee in one turn?AA SLAA HLAI SLAI HLSolid
- How many attempts until the rare item? Drop rates and pity systemsAA SLAA HLAI SLAI HLSolid
- How many different games of noughts and crosses are there?AA SLAA HLAI SLAI HLSolid
- How many shuffles does it take to mix a deck of cards?AI HLAA HLAI SLAmbitious
- What are the odds in a Risk battle?AA SLAA HLAI SLAI HLSolid
- Where should you build your first settlement in Catan?AA SLAI SLAA HLAI HLAccessible
- One snake queue or a queue at every till? Simulating supermarket checkoutsAI SLAI HLAA SLAA HLSolid
- How far does a random walker get? Distance after n stepsAA HLAA SLAI HLSolid
- Why do buses come in bunches? A simulation of a bus routeAI HLAI SLAA HLAmbitious
- How many random numbers until the total passes 1? Finding e by chanceAA HLAA SLSolid
- Does the best team win? Knockout cups versus leaguesAI SLAI HLAA SLAA HLSolid
- What is a 'random' chord? Bertrand's paradox simulatedAA HLAA SLSolid
- Finding the key length of a Vigenère cipher with the index of coincidenceAA HLAI HLAmbitious
- How long would it take to guess your password? Counting and logarithmsAA SLAI SLAA HLAI HLAccessible
- Why does a random game draw the Sierpiński triangle?AA SLAA HLAI SLSolid
- Does rain today predict rain tomorrow? A Markov chain for the weatherAI HLAA HLSolid
- Can letter transitions tell languages apart? Markov chains for textAI HLAA HLAI SLSolid
- Gambler's ruin: how long until someone runs out?AI HLAA HLAmbitious
- How likely is a voting paradox? Rock–paper–scissors preferencesAA HLAI HLAmbitious
- Left, right or centre? Game theory for penalty kicksAA SLAI SLAA HLAI HLSolid
- Secret Santa: how often does someone draw their own name?AA HLAA SLSolid
- How much should you bid in a sealed-bid auction?AA HLAA SLAmbitious
- Does the wording change the answer? A split-sample survey experimentAI SLAI HLAA SLSolid
- Getting honest answers: the randomised response techniqueAA HLAI HLAA SLAmbitious
- Counting a hidden population: capture–recapture at schoolAI SLAA SLAI HLAA HLSolid
Frequently asked questions
Can a probability IA get top marks without any data?
Yes. A theoretical investigation can score full marks if the question is clear, the reasoning is correct and well explained, and you reflect on the assumptions. Simulations or small experiments make the reflection much easier.
Is the birthday problem too simple for an IA?
On its own, yes. It becomes a good IA if you extend it — for example to near-matches (birthdays within a day), unequal birth rates across the year, or the expected group size for the first match.
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