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

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

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

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.

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

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

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.

Taking it further

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