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Subject: Re: Proving something is better

Author: Peter Fendrich

Date: 03:31:57 12/23/02

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On December 22, 2002 at 07:48:37, Rémi Coulom wrote:

>This is what my program says:
>
>number of wins   n1 = 4
>number of losses n0 = 6
>P(p1>p0) = 0.274414
>
>Below is what your own program says. 4 6 100 is only slightly different, but
>that is certainly related to the incaccuracies of the Monte Carlo method.
>
>Result Evaluator. Version 0.8
>by Jean-Peter Fendrich; peter@fendrich.net
>Enter 'Help' for more information
>
>
>Enter a match result in the form Wins Losses Draws or Q for quit. 4 6 0
>10 games. Result: 4.0-6.0
>Some different values of Pw and Pd:
>Pw      Pd      Trin            cumTrin
>------  ------- -----------     ---------
>0.33    0.33    0.00355636844   0.05450558
>0.40    0.20    0.022020096     0.122454
>0.49    0.01    0.189157533     0.7482712
>0.37    0.26    0.0100980272    0.07425042
>0.33    0.35    0.00276091416   0.05424455
>0.01    0.99    2.05078125e-021 0.9980402
>0.80    0.10    8.6016e-005     0.6260984
>............................................................................
>Probability for A better than B is: 0.2745533
>
>Enter a match result in the form Wins Losses Draws or Q for quit. 4 6 1
>11 games. Result: 4.5-6.5
>Some different values of Pw and Pd:
>Pw      Pd      Trin            cumTrin
>------  ------- -----------     ---------
>0.33    0.33    0.00118545615   0.03502741
>0.40    0.20    0.0044040192    0.0886891
>0.49    0.01    0.00189157533   0.6468857
>0.37    0.26    0.00262548707   0.05006582
>0.33    0.35    0.000966319957  0.03491507
>0.01    0.99    2.03027344e-021 0.9975906
>0.80    0.10    8.6016e-006     0.5771945
>............................................................................
>Probability for A better than B is: 0.2747703
>
>Enter a match result in the form Wins Losses Draws or Q for quit. 4 6 10
>20 games. Result: 9.0-11.0
>Some different values of Pw and Pd:
>Pw      Pd      Trin            cumTrin
>------  ------- -----------     ---------
>0.33    0.33    9.56946654e-008 0.0008574022
>0.40    0.20    3.58271855e-009 0.01094223
>0.49    0.01    3.00550302e-021 0.6054174
>0.37    0.26    2.26497568e-008 0.002691502
>0.33    0.35    1.21011565e-007 0.0008517853
>0.01    0.99    2.94689602e-021 0.9916248
>0.80    0.10    1.36669867e-014 0.2530475
>............................................................................
>Probability for A better than B is: 0.274245
>
>Enter a match result in the form Wins Losses Draws or Q for quit. 4 6 100
>110 games. Result: 54.0-56.0
>Some different values of Pw and Pd:
>Pw      Pd      Trin            cumTrin
>------  ------- -----------     ---------
>0.33    0.33    2.33563783e-045 1.125915e-019
>0.40    0.20    9.44803171e-067 1.700053e-011
>0.49    0.01    6.40245019e-196 0.1909135
>0.37    0.26    1.07422165e-055 3.536888e-015
>0.33    0.35    2.38440436e-043 2.734289e-019
>0.01    0.99    2.54074667e-016 0.8122032
>0.80    0.10    2.91139955e-099 2.06995e-005
>............................................................................
>Probability for A better than B is: 0.2773552
>
>Enter a match result in the form Wins Losses Draws or Q for quit. Q
>Result Evaluator ended

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

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.

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



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