📄 buffon's needle.htm
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<H2><CENTER>Buffon's Needle<BR>
An Analysis and Simulation</CENTER></H2>
<CENTER>by<BR>
<A HREF="../reese.html">George Reese</A>
<P><IMG SRC="display.gif" WIDTH=224 HEIGHT=133 X-CLARIS-USEIMAGEWIDTH X-CLARIS-USEIMAGEHEIGHT ALIGN=bottom></P></CENTER>
<P>
<HR>
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<UL>
<LI><A HREF="bufjava.html">Run the simulation using a Java
applet</A></LI>
<LI><A HREF="#intro">Introduction</A> to the problem.</LI>
<LI><A HREF="#simple">The simplest case</A></LI>
<LI><A HREF="#othercases">The other cases</A></LI>
<LI><A HREF="close.html">An estimate that is very close</A></LI>
<LI><A HREF="#questions">Questions</A></LI>
<LI><A HREF="#references">References</A></LI>
<LI><A HREF="buffonneedle.sea.hqx">Download a graphical simulation
program</A> (Macintosh, binhexed)
<LI><A HREF="reese-buf.jpg">An interesting extension</A> posed by p. ganguly.
<BR><a href="/contact?to=George Reese">Send me an email</A> if you think you have an answer.
<P> </P></LI>
</UL>
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<HR>
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<H2><A NAME=intro></A>Introduction</H2>
<P>Buffon's Needle is one of the oldest problems in the field of
geometrical probability. It was first stated in 1777. It involves
dropping a needle on a lined sheet of paper and determining the
probability of the needle crossing one of the lines on the page. The
remarkable result is that the probability is directly related to the
value of pi.</P>
<P>These pages will present an analytical solution to the problem
along with a Java program for simulating
the needle drop in the simplest case scenario in which the length of
the needle is the same as the distance between the lines.</P>
<P> </P>
<H2><A NAME='simple"'></A>The Simplest Case</H2>
<P>Let's take the simple case first. In this case, the length of the
needle is one unit and the distance between the lines is also one
unit. There are two variables, the angle at which the needle falls
(theta) and the distance from the center of the needle to the closest
line (D). Theta can vary from 0 to 180 degrees and is measured
against a line parallel to the lines on the paper. The distance from
the center to the closest line can never be more that half the
distance between the lines. The graph below depicts this
situation.</P>
<CENTER><IMG SRC="situation.gif" WIDTH=318 HEIGHT=130 X-CLARIS-USEIMAGEWIDTH X-CLARIS-USEIMAGEHEIGHT ALIGN=bottom></CENTER>
<P>The needle in the picture misses the line. The needle will hit the
line if the closest distance to a line (D) is less than or equal to
1/2 times the sine of theta. That is, D <= (1/2)sin(theta). How
often will this occur?</P>
<P>In the graph below, we plot D along the ordinate and
(1/2)sine(theta) along the abscissa. The values on or below the curve
represent a hit (D <= (1/2)sin(theta)). Thus, the probability of a
success it the ratio shaded area to the entire rectangle. What is
this to value?</P>
<CENTER><IMG SRC="function.gif" WIDTH=352 HEIGHT=122 X-CLARIS-USEIMAGEWIDTH X-CLARIS-USEIMAGEHEIGHT ALIGN=bottom></CENTER>
<P>The shaded portion is found with using the definite integral of
(1/2)sin(theta) evaluated from zero to pi. The result is that the
shaded portion has a value of 1. The value of the entire rectangle is
(1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 2/pi.
That's approximately .6366197.</P>
<P>To calculate pi from the needle drops, simply take the number of
drops and multiply it by two, then divide by the number of hits,
or<BR>
2(total drops)/(number of hits) = pi (approximately).</P>
<P> </P>
<H2><A NAME=othercases></A>The Other Cases</H2>
<P>There are two other possibilities for the relationship between the
length of the needles and the distance between the lines. A good
discussion of these can be found in <A HREF="#schroeder">Schroeder</A>,
1974. The situation in which the distance between the lines is
greater than the length of the needle is an extension of the above
explanation and the probability of a hit is 2(L)/(K)pi where L is the
length of the needle and K is the distance between the lines. The
situation in which the needle is longer than the distance between the
lines leads to a more complicated result.</P>
<P> </P>
<H2><A NAME=questions></A>Questions</H2>
<UL>
<LI><B>1.</B> After 1,000 drops, how close would you expect to be
to pi?
<P><B>2.</B> After 264 drops, the estimate of pi is 3.142857. This
estimate is<BR>
correct to within 2/1000 of the book value of pi. Will the next
drop:<BR>
</P>
<UL>
<LI>A. make the estimate more accurate?<BR>
B. make the estimate less accurate?<BR>
C. make it more or less accurate depending on whether it's a
hit or miss? or<BR>
D. impossible to say.</LI>
</UL>
<P><B>3.</B> What about the next 10 drops?<BR>
</P></LI>
</UL>
<H2><A NAME=references></A>References</H2>
<P>Cheney, W. and Kincaid, D. (1985). <B>Numerical Mathematics and
Computing.</B> 2nd Ed. Pace Grove, California: Brooks/Cole Publishing
Company pp. 354-354</P>
<P><A NAME=schroeder></A>Schroeder, L. (1974). Buffon's needle
problem: An exciting application of many mathematical concepts.
<B>Mathematics Teacher, 67</B> (2), 183-186.</P>
<P></P>
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