Buffon Laplace Needle Problem.

[Patrick D. Rockwell]

I found out that there is a Buffon Laplace Needle Problem which says that
if you drop a needle of length l on a grid of squares who's dimensions are
aXb
then the probability of the needle crossing a line is


(2*l*((a+b)-l^2)/(a*b) if l<a,b

but what about the cases where l>=a,b?

I believe that as long as the length of the needle is
<=(a^2+b^2)^0.5 then there is still a possibility
that it won't cross a line.

What about the 3 dimensional case, or the 4 dimensional
case where you have a matrix of cubes who'se dimensions are
aXbXcX.... and a needle of length l. how would you do something
like that?

Thanks in advance for any help.

[Stephen J. Herschkorn]

Post by Patrick D. Rockwell
I found out that there is a Buffon Laplace Needle Problem which says that
if you drop a needle of length l on a grid of squares who's dimensions are
aXb
then the probability of the needle crossing a line is
(2*l*((a+b)-l^2)/(a*b) if l<a,b
but what about the cases where l>=a,b?
I believe that as long as the length of the needle is
<=(a^2+b^2)^0.5 then there is still a possibility
that it won't cross a line.
What about the 3 dimensional case, or the 4 dimensional
case where you have a matrix of cubes who'se dimensions are
aXbXcX.... and a needle of length l. how would you do something
like that?

I am most used to the the two-dimensional case where there are only
parallel lines, not a grid. Still, The approach is the same: Define
your random variables, set up the integral of the appropriate region. I
suspect that the case where ell >= min(a,b) is a messier expression.

I continue to find it pretty neat that 2/pi is the answer for parallel
lines with needle length equal to the distance for parallel lines.

BTW, best not to use an ell as a variable name, especially in posts. In
writing, at least we can make it a script ell.

[Jim Ward]

Didn't someone show that Buffon cheated, he kept dropping until he got
a great approximation for pi, then quit?

[Bill Taylor]

There's a wonderful proof of the Buffon needle problem based on that most
extraordinarily useful lemma for a bunch of random variables (not nec indpt):

the expectation of the sum is the sum of the expectations.


For each infinitesimal part of the thrown needle, the expectation of
the number of times it crosses (0 or 1) is the probability of crossing,
so adding them all up:-

the mean total number of crossings is proportional to the length.

This is true whether it's a needle that's thrown or any other finite bit
of a curve - as long as it has given length! Even better - it doesn't have
to be rigid - a piece of spaghetti would do!!

Finally, if one throws a rigid circle-shaped piece of wire, a ring, whose
diameter is the same as the grid's inter-line distance, then the actual
number of crossings is exactly two, thus so is the expected number,
from which you can get the constant of proportionality, and thus
solve Buffon without any calculus or whatever!!

Cool cool cool.


Incidentally, for a grid in two directions, wire-netting style, one can use
the basic lemma again and just add up the two expected numbers to get
the total expected number (though not the probability in general).

[Robert Israel]

Post by Bill Taylor
This is true whether it's a needle that's thrown or any other finite bit
of a curve - as long as it has given length! Even better - it doesn't have
to be rigid - a piece of spaghetti would do!!

Yes, that's called "Buffon's noodle".

Robert Israel ***@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada