IA idea · Simulation & Monte Carlo methods

How many random numbers until the total passes 1? Finding e by chance

AA HLAA SL Solid Also in: Pure maths, Probability

Research question

If you add random numbers between 0 and 1 until the total exceeds 1, how many do you need on average, and why is the answer e?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A spreadsheet suggests the answer quickly, which turns the IA into a hunt for the reason. The proof needs probability, a geometric argument about volumes and a series for e, all within reach of a strong AA student.

The mathematics you'll need

Simulation with RAND()
P(sum of n uniforms < 1) = 1/n!, from areas and volumes
Expected value as a series
The series for e (HL: Maclaurin series)
Standard error of the simulated estimate

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

No data needed: simulate thousands of trials in a spreadsheet.

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

Simulate and estimate the mean.
Find P(N > 2) from the area of a triangle; P(N > 3) from a volume.
Conjecture and justify P(N > n) = 1/n!.
Sum the series to get e.
Compare with the simulation and its error.

Pitfalls that cost marks

Stopping at the simulation.
Asserting 1/n! without a geometric or inductive argument.
Not quantifying the simulation's uncertainty.

Showing personal engagement

Predict the answer before simulating.
Change the target from 1 to 2 and investigate.
Use dice instead of random numbers and compare.

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	Simulation with RAND; P(sum of n uniforms < 1) = 1/n!, from areas and volumes
AA HL	Good fit	Simulation with RAND; P(sum of n uniforms < 1) = 1/n!, from areas and volumes
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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

Personal engagement (C)

Design the simulation yourself: choose the rules, test them on a small case you can check by hand, and change one rule at a time to answer a question you care about.

Reflection (D)

Compare simulation with exact theory or real data, say how many runs you used and how much the answer varies between batches, and question the random-number assumptions. For this idea, start with: stopping at the simulation — say how it affects your answer.

Use of mathematics (E)

SL: A probability model described precisely, simulated correctly, with the simulated answer compared with an exact calculation for at least one simple case and the results summarised with appropriate statistics.

HL: An estimate of the simulation's error (standard error, or a confidence interval for the estimate), a distribution fitted to the results and tested, or an exact result proved for the general case that the simulation confirms.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Find the expected number needed to pass a general target t, or the distribution of the final total.

Extending it for HL

Give a confidence interval for each simulated estimate and show how it narrows as the number of runs grows, or prove a general result that the simulation confirms.

See a complete IA, marked

Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) asks a different question, but shows how a complete simulation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).