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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

499.0. "Buffon buffoons" by METOO::YARBROUGH () Wed Jun 04 1986 13:49

Albert and Robert were repeating, for lack of anything better to do, 
Buffon's Needle Experiment: Estimating Pi by dropping a needle of length X
on a floor marked with parallel lines separated by distance X, counting the
times when the needle landed across a line and comparing with the count of
times when the needle fell between lines. They had been doing this all
evening, running up nearly 100,000 trials, and both were nearly exhausted.
They had reached a point where their last estimate gave Pi to seven
significant figures, but Alfred wasn't satisfied. 

"Let's keep on for another thousand trials, Robert", said Alfred. "I want
to see if we can get another significant digit for Pi." 

"No way, I'm bored out of my mind," said Robert. "I'm not going to stick
around. The chances of our getting any improvement are too slim." 

"O.K, I'm going to try another 1000 trials, then I'll quit, unless I get a 
better estimate along the way. I'll let you know what I have found in the 
morning."

What are Alfred's chances of geting an improvement in his estimate in 
another 1000 trials? With 2000 trials? With 4000?
T.RTitleUserPersonal
Name
DateLines
499.1ENGINE::ROTHThu Jun 05 1986 02:293
    Must have been a pretty accurately measured needle and floor!

    - Jim
499.2More PrecisionVAXRT::BRIDGEWATERSat Jun 07 1986 22:2922
    I am a little confused as to exactly what you consider an improved
    estimate of pi for the purposes of this problem.
    
    1.  Do you mean that an improved estimate of pi would require only
        that the new estimate be closer to pi than Albert's and Robert's
        first estimate?
    
    2.  Or, do you mean that it would require them to obtain the first
        eight digits of pi as they appear in the infinite precision decimal
        expansion of pi when their estimate is rounded to the nearest
        8-digit decimal number?  (3.1415926)
    
    3.  Or, do you mean that when pi and their estimate are both rounded
        to the nearest 8-digit decimal number that these resulting two
        numbers are identical?   (3.1415927)

    A possible partial answer follows:

    If #3 is the meaning of an improved estimate of pi, then it is not
    possible to get a better estimate of pi in up to 2927 extra tries.
    
    - Don
499.3Educating AlbertMODEL::YARBROUGHThu Aug 07 1986 15:459
    The key to understanding this problem is to realize that Buffon's
    experiment yields a rational approximation to Pi, i.e.
    	<trials>/<successes>.
    To get seven correct digits, the last set of results must have been
    some multiple of the well-known 355/113, because the next more accurate
    rational approx. is 103993/33102. The largest multiple of 355 under
    100000 is 99755, which means Al has at least another 4238 trials to go to
    have a chance of improvement. So the probability of improvement
    in 4000 trials is zero.
499.4CLT::GILBERTeager like a childThu Aug 07 1986 17:5619
I'm lost.  Could you explain this again?  Note that 52163/16604 is a
better approximation to pi than 355/113.


The fact that pi is involved seems irrelevant.  We have:

	| X     |     -7           5
	| - - p | < 10  ,  X+Y < 10 , 0 < X, 0 < Y, p is irrational
	| Y     |

And we would like to prove that

	| X+x     |   | X     |
	| --- - p | < | - - p |, 0 < x, 0 < y
	| Y+y     |   | Y     |

implies
	        3
	x+y > 10
499.5luck enters into the experimentMODEL::YARBROUGHFri Aug 08 1986 15:4114
    Among all the rational approximations to Pi that you might encounter
    in Buffon's experiment are a few very good ones (multiples of 355/113,
    52163/16604, etc.), and a lot of bad ones. Along the way you may
    or may not meet any of the good ones. In particular, in the vicinity
    of 99,000 - 100,000 the only good ones are the multiples of 355/113.
    When you get beyond that point the next ones you meet that are not
    such a multiple are the two I cited and also 2*52163/2*16604. It
    looks like Albert's best bet (after the 10,000 trials of the next
    problem) is to run into 3*52163/3*16604, although while that is a more 
    accurate approximation to pi it does not provide the extra digit of 
    precision Albert was seeking.
    
    It's possible I have missed another more accurate RatApprox somewhere
    along the line - if so I will have to revise my problem!
499.6CLT::GILBERTeager like a childFri Aug 08 1986 17:126
	a*52163+b*355					    355
   But	-------------  is a better approximation to pi than ---,
	a*16604+b*113					    113

  for any positive a and b.  But there should be a simple explanation
  of why they can't get another 'digit'.