Probability · Maths EE idea · Solid
Buffon's needle and estimating π
A research question to start from
Why does the probability that a dropped needle crosses a line involve π, and how accurate is the resulting estimate of π for a given number of drops?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Geometric probability with an exact integral and a statistical analysis of the estimator.
Mathematics you would need
- Continuous random variables
- Double integrals over a region
- Expected value and variance of an estimator
- Normal approximation
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Derive the crossing probability for short needles.
- Extend to long needles.
- Analyse the estimator's variance and how many drops a given accuracy needs.
Scope and difficulty
Solid. Solid.
Pitfalls
- Only an experiment.
- No error analysis.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Buffon's needle derivation long needle; variance of Buffon estimator. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).