9. On a slightly less satirical note...
Posted by: "dixiepokerace" bigrich@publicist.com dixiepokerace
Date: Wed May 13, 2009 12:18 pm ((PDT))I've been following this whole "gaffed chip" sequence with my usual mote of
interest, and although I thoroughly enjoy lampooning the tin hat brigade, I do
feel that the mathematically inclined among us are guilty of a little blindness
as well.For instance, if someone says that he/she lost 20 hands in a row playing 9/6
JoB, the response would properly be "tough luck". But what if it were 200 hands
in a row? 2000? Certainly even though those numbers are "too small a sample"
for ordinary analysis, there must be some threshold where even a very small
sample can yield a very suspicious result, no? I am formerly a math whiz, but
haven't exercised my stats muscles (ok, ANY muscles) in decades, so maybe
someone else out there can refresh my memory on standard
deviation/variance/Chi-Square or whatever it is called? I think I remember that
six standard deviations (Six Sigma, as the business folk like to call it) is a
significant boundary in that should a result fall outside of that range it would
be considered "anomalous"? How many hands lost in a row WOULD be cause for
concern?Just wonderin'....
I'm in the same boat as you regarding recent exercise of my statistical and other muscles, also with good exposure to the topic (both of them) back in college.
I therefore can no longer give you the exact numbers, while others can (and probably will) do so. But it's more important to me just to remember the underlying concept, regardless of what the numbers actually are.
The key to remember is that the bigger the sample, the higher the probability that the sample reflects the larger population of which it is a sample (assuming it is a random sample, reasonable for VP machines, but not always so when doing surveys).
Likewise, if one is using the sample to "test" something for compliance with a predicted frequency, which is what most of these posts seem to be doing, the larger the sample, the higher the probability that one can say that the observed frequency does (or does not) reflect variation from the predicted frequency BY CHANCE ALONE.
So, to use the original example, 20 flush draws with narry a hit is a sample to which one can assign a number, using the statistics, which will tell you that there is only a (for example) probability of 0.01 (one chance in a hundred) that this will occur BY CHANCE ALONE. BUT - there IS still that small chance that it occurs by chance alone, and does not represent a gaffed machine.
So (and again, my numbers are made up), with that 1% chance, 100 of us go out and play for a while and have 20 flush draws. Even on non-gaffed machines, one of us will usually miss all 20 draws -- even if we all play the same machine!
Again using hypothetical figures (the mathematicians will provide the correct ones), if one has 1,000 hands in a sample, one might be able to say with 90% certainty that the number of two-pair hands in the sample is within X% of the number you would EXPECT to get if you played an infinite number of hands. If the sample goes to 10,000 hands, the percentage of certainty might go up to 95%, or 99%. It never reaches 100%.
Conversely, these percentages mean that there is a 10% and 5% chance, respectively, that the number has varied from the predicted value by chance alone, the predicted value being that which is calculated assuming a "true" deck and a truly random selection of hands.
This is where people get into the idea that a machine is "gaffed" - they "know" that by calculation, they "should" get XX hands of a certain kind in their small sample. If they get significantly fewer than XX ("significantly" being subjective), they assume the machine is not dealing "fairly" or "randomly". In fact, no matter how large their sample, there is always a small probability that it will deviate "significantly" from predicted frequency by chance alone -- the larger the sample, the smaller that probability, but it's always there.
Statistics allows us to say things with xx% certainty, but xx is never 100%!
The simplest example is the coin flip. On a "true" or "fair" coin, after 10 flips, there is about one chance in 1,000 that it will have come up heads (or tails, if you choose) every time, and in fact, one chance in 500 (these are rough figures, but NOT made up) that one of the two "always" events will occur. If you had to evaluate whether you thought the coin was "fair", you MIGHT want to say "no" after 1,000 flips if you got all heads or all tails. The odds are about 500 to 1 that you'd be right, but you can never say it with absolute (i.e. 100%, no rounding, exactly 100%) certainty based only on statistical sampling.
VP is much more complicated because there are so many more possible events, and each of us tends to focus on the one that troubles them the most (even knowing all this, I also "have trouble" making flushes, and I "know" that I rarely improve a pair of 7's, but have a better than usual improvement if I get to hold three 7's). Each of us struggles with our natural tendency to "learn from experience" and has to fight what we "know" should happen when it does not in fact occur!
And of course, some of the problem comes, as occurred with the original post, of someone thinking (for whatever reason) that "20" hands is an "almost infinite" sample! Infinity is much larger than 20 as I recall. Again, the mathematicians will be able to tell you exactly how much larger 
--BG
···
===================