| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 499.1 | | ENGINE::ROTH | | Wed Jun 04 1986 23:29 | 3 |
| Must have been a pretty accurately measured needle and floor!
- Jim
|
| 499.2 | More Precision | VAXRT::BRIDGEWATER | | Sat Jun 07 1986 19:29 | 22 |
| 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
|
| 499.3 | Educating Albert | MODEL::YARBROUGH | | Thu Aug 07 1986 12:45 | 9 |
| 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.
|
| 499.4 | | CLT::GILBERT | eager like a child | Thu Aug 07 1986 14:56 | 19 |
| 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
|
| 499.5 | luck enters into the experiment | MODEL::YARBROUGH | | Fri Aug 08 1986 12:41 | 14 |
| 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!
|
| 499.6 | | CLT::GILBERT | eager like a child | Fri Aug 08 1986 14:12 | 6 |
| 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'.
|