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A Problem in Geometric Probability: Buffon’s Needle Problem

A Problem in Geometric Probability: Buffon’s Needle Problem. The Plan. Introduction to problem Some simple ideas from probability Set up the problem Find solution An approximation Generalization (solution known) Other generalizations ( solutions known?).

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A Problem in Geometric Probability: Buffon’s Needle Problem

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  1. A Problem in Geometric Probability: Buffon’s Needle Problem

  2. The Plan • Introduction to problem • Some simple ideas from probability • Set up the problem • Find solution • An approximation • Generalization (solution known) • Other generalizations ( solutions known?)

  3. Buffon's Needle ProblemStated in 1733 solution published 1777by Geroges Louis Leclerc, Comte de Buffon (1707-1788)

  4. P(landing on red) = P(landing on c) =

  5. Southern end = measuring end D d Measure to the closest line north of the measuring end We have a crossing if y q L The Set Up

  6. The Solution!

  7. Polyá, George (1887, 1985) … a good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. How to Solve It. Princeton: Princeton University Press. 1945.

  8. We also know that Something different Let L=1 and D=4, then we have Simulation

  9. Generalizations

  10. D L E

  11. Other Generalizations

  12. P(crossing) = P(crossing) Total number of throws Any Uses? Counting white blood cells!

  13. MATLAB CODE function p=buffon(L,D,n) cnt=0; for i=1:n x=rand*(pi/2); y=rand*D; if y <= (L*sin(x)) cnt=cnt+1; end end p=cnt/n;

  14. OUTPUT EDU»pi ans=3.14159265 EDU» buffon(1,4,100) ans=2.9412 EDU» buffon(1,4,1000) ans= 2.8409 EDU» buffon(1,4,3000) ans= 3.1646 EDU» buffon(1,4,10000) ans= 3.1586 EDU» buffon(1,4,100000) ans= 3.1342

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