Sorry, I accidentally sent the last post before I intended. Here is the rest.
The probabilities for a normal distribution are actually a little worse than Robert suggested
For +- 3 st dev and a truly normal distribution, odds are indeed 0.0027.
For +-2 st dev, odds of failure are 0.0455 (which is often rounded to 0.05) – 0.0027 (you can throw out cases where you are beyond +-3 st dev because the first rule already catches these) = 0.042. The odds of 2 out of three failing are
(0.042 x 0.042) x (1-0.042) x 3 = 0.0052
(2 failures) x (one success) x (three locations for the succes)
For +-1 st dev, 0.317 are outside the limits. Odds of 4 out of 5 outside the limits are
(0.317 x 0.317 x 0.317 x 0.317) x (1-0.317) x 5 = 0.034
Thus, from any given point, the odds of a specfic rule coming immediately into effect are quite different.
Tim
P.S. Even this is too simple to determine the real odds of a rule coming into play, for a few reasons.
1) real distributions are not exactly normal.
2) the number of opportunities for each type of occurance change. For example, if you do 5 runs, there is only one opportunity of getting 4 out of 5, but there are 5 opportunities for getting 1 out of 1 (= 0.013 odds of failure for the 5 runs).
3) The multi-trial rules are self-correlated. For example, if 2 in a row are outside +-2, then there are two bad runs, (including either the good one before or after) but you only count 1 failure.
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