A Problem in Geometric Probability: Buffon’s Needle Problem

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Presentation transcript:

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?)

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

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

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

The Solution!

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.

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

Generalizations

D L E

Other Generalizations

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

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 p=cnt/n;

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