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Subject: Re: Proving something is better

Author: Rémi Coulom

Date: 06:10:59 12/23/02

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On December 23, 2002 at 06:31:57, Peter Fendrich wrote:

>
>Remi,
>I don't get the same results as you:
>(all the sample values of Pw, Pd, Trin and cumTrin are the same)
>          Mine         Yours
>4 6 0 :   0.2745363   0.2745533
>4 6 1 :   0.274543    0.2747703
>4 6 10:   0.2740947   0.2773552
>
>I'm not sure what's happening here but as you say, the Monte Carlo method
>doesn't give exact values.
>I thought that the program could be reliable to 3 decimals but maybe not...
>However, if I'm right about draws, the prob would slowly move towards 0.5 when
>the number of draws increases. I continued up to 500 draws and got the
>following:
>4 6 100:  0.2758096
>4 6 200:  0.2820387
>4 6 300:  0.2906555
>4 6 400:  0.3076096
>4 6 500:  0.3273436

That is because the Monte Carlo method is inaccurate: think about the x^n
function, x varying between 0 and 1: when n grows large, the function has an
extremely thin peak, that is very difficult to integrate accurately with a Monte
Carlo method.

I am totally certain about this, because I first started by implementing a
program that calculated the big trinomial integrals. I wrote a small polynom
library based on the GNU multiprecision library so that the _exact_ value was
found. That's how I noticed that the result does not depend on draws, and went
further in the calculations.

>
>The probability for A neatly grows with more draws.
>OTOH I can't argue against your formulas. Give me some more time.
>Could you please elaborate the first formula in section "3 Draws do nout Count".
>How do explain 1 - p0 - p0.5 = 1 - u + up0.5 - p0.5?
>I'm a bit interested in the term up0.5 that seems to be superfluous.

u is defined by p0 = u * (1 - p0.5). If you replace p0 by u(1 - p0.5) in
1 - p0 - p0.5, the you get 1 - u + up0.5 - p0.5

>
>Maybe we should continue via email if no one else shows interest in the
>discussions.
>I don't know if Uri still follows it, if so shout...
>
>/Peter

We are at the very bottom of the message board now, so I suppose we are not
annoying anybody.

Rémi



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