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Binomial Distribution
Many MMO events are two-state systems, also known as Binomial. Anything that either can happen or cannot (Zanshin, Double Attack, Hit Rate, Magic Critical Hit, etc.) are all binomial when framed correctly.
For instance, it's widely accepted that Triple Attack procs before Double Attack. Therefore, a THF/WAR cannot take their raw Double Attack rate and assume it's binomial. There were not two options, there were actually three (TA, DA, and single attack). However, the same player could make their Double Attack rate binomial by subtracting out rounds where Triple Attack procced. With Triple Attack removed, only two options remain (Double Attack and single attack).
Though the reasoning is beyond the scope of this article, things that are binomial have very nice, simple statistics. They can be easily used to calculate a confidence interval with a normal approximation (valid when n×p>5 and n×(1-p)>5):
- 95% Conf. Interval Width = 1.96*SquareRoot(p×(1-p)÷n)
In the above equation, p is the probability of something occurring from the sample you have collected. So if you have a 15.05% Double Attack rate on your WAR/NIN, p=0.1505 . Also, n is the sample size. So if your 15.05% Double Attack rate was the result of 45 Double Attacks / 299 Attack Rounds, your n=299. All told, the above equation gives you 15.05±4.05% for your confidence interval.
In a more complicated case, where you need to (for instance) filter out Triple Attack, the parse would say it like this:
- Triple Attack rate = 15.05% = 45/299
- Double Attack rate = 11.7% = 35/299
- However, p = 35÷(299-45) = 13.78%, because Double Attack was unable to proc on 45 of the attack rounds.
This may change in a future parser version, but at the moment this is a correction that you have to make for yourself.
The above confidence interval assumes normality, which may not be valid (could underestimate the confidence interval) when near 0% or 100%. A more conservative way to estimate the confidence interval (the Clopper-Pearson method) is linked below, and should be used when you have low N or a probability near 0% or 100%.
