Re: Law of Independent Trials: Is It Flawed?



Posted by Bryce Carlson on June 26, 1997 at 19:52:49: In Reply to: Law of Independent Trials: Is It Flawed? posted by NoMath on June 26, 1997 at 12:44:05:

Yes, you are, indeed, missing something, NoMath. To begin with, 5 to 1 odds are, in fact, "fair" odds for your proposition, in that they convey no edge to either side. But that aside, what I think you are missing is that, although as the number of trials increases the *percentage* of, say, 7s rolled converges on 1 in 6, the absolute numeric divergence of 7s from exactly 1 in 6 actually *increases*. That is, irrespective of the number of trials, the most likely outcome (the median result) is 1 in 6, but as the number of trials approaches infinity, the possible numeric deviations from this number (1 in 6) also approaches infinity. Intuitively, this makes sense: as the number of trials increases, the deviant results far out on the left or right of the normal-distribution "tail" increase in absolute numbers, but decrease as a percentage of total trials. So, the commonly-held belief that the "law of averages" demands that as the number of trials approaches infinity, the result must asymptotically approach the true odds in absolute numeric terms (in this case, occur with a frequency of exactly 1 in 6) is dead wrong. In actuality, as the number trials increases, the deviation in absolute numeric terms from the true odds also increases (though as a percentage it does approach the true odds).

Bryce Carlson


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