Re: Theory: 11.7447%, Excel Simulation: 10.4%, Mathematica Sim 11.3%
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I reran the Excel simulation with your corrections. Among the 200,000 virtual deals there were 20,789 deals that had 14 pairs or more in a window. That simulation gives a 10.4% estimate of the probability.
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Just so we are clear, does 200k deals mean 200 million hands?
In any case, I also wrote a sim from scratch in Python. After 20k trials (i.e. up to 20 million "hands"), I got around 10.3%, which has a conf. interval size around +/- half a percent, so I believe it is below 11%.
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I wrote a simulator in Mathematica and got 22,545 deals with 14 pairs or more. That's 11.3%. Given the size of the simulations, either there is a bug one of the simulation programs and the theoretical derivation, or these random generators are not good enough. Hmm.
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It is not a problem with the random generators (though with something that big, I wouldn't trust Excel completely). Perhaps we both made the same sim mistake, whatever it is. I also checked some of your numbers; I agree with your "r" values, etc. We are missing something. Oh well.
alThor
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