Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Estimating π by dropping needles: how fast does the estimate improve?

AA HLAA SLAI SLAI HL Solid Also in: Pure maths, Calculus, Simulation

Research question

How does the accuracy of Buffon's needle estimate of π improve with the number of drops, and is it consistent with the standard error predicted by the binomial distribution?

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Why it makes a good exploration

Dropping sticks on floorboards to estimate π is a classic, but the interesting mathematics is in the error: how many drops do you need for 3 correct decimal places?

The mathematics you'll need

Geometric probability
Integration to derive P(cross) = 2l/(πd)
Binomial variance and standard error
Simulation and convergence plots

Where the data comes from

Drop matchsticks on lined paper (hundreds of times) and supplement with a simulation.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Derive the crossing probability.
Collect physical drops; estimate π.
Simulate up to 10⁶ drops; plot error against drops.
Compare with the predicted standard error.
Reflect on why this is a poor way to compute π and why it matters anyway.

Pitfalls that cost marks

Deriving with a hand-wave instead of an integral.
Needle longer than line spacing (different formula).
No error analysis.

Showing personal engagement

Do it with your class and pool the results.
Try different needle lengths and predict the best ratio.
Discuss Lazzarini's suspiciously accurate 1901 result.

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	Geometric probability; Integration to derive P(cross) = 2l/(πd)
AA HL	Good fit	Geometric probability; Integration to derive P(cross) = 2l/(πd)
AI SL	Good fit	Geometric probability; Integration to derive P(cross) = 2l/(πd)
AI HL	Good fit	Geometric probability; Integration to derive P(cross) = 2l/(πd)

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: deriving with a hand-wave instead of an integral; and needle longer than line spacing (different formula). Say how each one affects your answer.

Use of mathematics (E)

SL: geometric probability; integration to derive P(cross) = 2l/(πd); binomial variance and standard error; simulation and convergence plots, each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Investigate the long-needle case or Buffon's noodle.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL probability 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 probability exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL, AA SL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Geometric probability, integration to derive P(cross) = 2l/(πd), binomial variance and standard error and simulation and convergence plots.

Where can I get data for this IA?

Drop matchsticks on lined paper (hundreds of times) and supplement with a simulation. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Estimating π by dropping needles: how fast does the estimate improve?” filled in: the question, the maths and the data sources.

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