Math 2.0

Probability theory and stochastic processes Combinatorial and geometric probability Problem

Buffon's needle problem

Builds on Geometric probability

Problem

A floor has parallel lines at distance t>0t > 0 apart. A needle of length ll with 0<l≤t0 < l \le t falls at random on the floor. "At random" means this. The distance xx of the centre of the needle from the nearest line is uniform on [0,t/2][0, t/2]. The acute angle θ\theta between the needle and the lines is uniform on [0,π/2][0, \pi/2]. The variables xx and θ\theta are independent. Find the probability that the needle crosses a line.

Remarks

The answer is 2lπt\dfrac{2l}{\pi t}. The needle crosses a line exactly when x≤l2sin⁡θx \le \frac{l}{2} \sin \theta, and the integral of this region gives the answer. For a needle with l=tl = t, the probability is 2/π≈0.6372/\pi \approx 0.637. The answer contains π\pi, so repeated throws give a random estimate of π\pi. For a long needle with l>tl > t, the answer is a different formula.

History

Georges-Louis Leclerc, Comte de Buffon, posed the problem in the 18th century. It was one of the first problems of geometric probability.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionGeometric probability, concept, no evidence levelGeometric probabilityconceptBuffon's needle problem, problem, no evidence levelBuffon's needle probl…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. Sample space definition
  15. The rationals form a field statement
  16. Universal quantifier definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Power set definition
  20. Universal set definition
  21. Order on the reals definition
  22. Complement definition
  23. Interval definition
  24. Sigma-algebra definition
  25. Event definition
  26. Probability axioms concept
  27. Probability measure definition
  28. Probability space definition
  29. Geometric probability concept
  30. Buffon's needle problem problem

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Buffon's_needle_problem, Solution
About this page
Tag06X5PF6YFP
URIm2://06X5PF6YFP
Aliasesbuffons-needle
Kindproblem
WritingW1 stated
MSC60D05
StatusThis copy of the site has no build data. Run m2 status 06X5PF6YFP.
UsesGeometric probability
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