Random is everywhere in WoW. Most spells have a range of damage. All weapons have a range of damage. Did you hit or miss? If you hit, was it a crit? Did your talents proc? How about your trinkets or weapon enchants? Maybe your embroidery?
In each action, a multitude of things can happen or not happen entirely randomly. When we talk about increasing DPS, increasing healing, or increasing survivability, we invariably talk about the midpoint, the average DPS something will grant. Moving from 20% crit to 26% crit is generally an increase of about 5% total damage (126/120). But if at the end of a fight you cast precisely 100 spells and 40 crit, how powerful was that increase to your crit chance? And what if only 15% crit?
So, this led me to a question. If we look at the *average* damage a player will do, what are the odds that he'll spontaneously do 10% more than normal? I'm going to be dealing only with crits here. Proc chances and things are beyond the scope of this project.
Lets normalize. We're casting 100 spells, each spell does 10 damage. A crit is 20 damage. At 30% crit, we're expecting 1300 total damage. This equates to 70 hits and 30 crits. A ten percent increase would be 1430 total damage, or 57 hits and 43 crits, or 43% crit chance.
The odds that we get 30 crits within 100 attacks is given by (CritChance^(NumCrits)) * ((1-CritChance)^(TotalAttacks-NumCrits)) * (Number of possible combinations of NumCrits and (TotalAttacks-NumCrits)) For example, given 2 coin flips of a fair coin, the odds of getting one head and one tails is .5^1 * .5^1 * 2 ([HT] or [TH] are both viable) which is .5 * .5 * 2 or .25 * 2 or .5, which makes sense, as there are four total options and two lead to the desired result. So, that math definitely works, and I feel better.
Crit Chance, Num Crits, and Total Attacks are all determined by our calculation. However, the number of possible combinations of crit/hit is harder to get at since we're dealing with reasonably large sample sizes (100). This is actually given to us by the formula [(Total Attacks)! / ([Num Crits]! * [Total Attacks - Num Crits]!)] or in our case 100! / (30! * 70!)
Thus, our whole formula is
(CritChance^(NumCrits)) * ((1-CritChance)^(TotalAttacks-NumCrits)) * ([TotalAttacks]! / [(NumCrits)! * (TotalAttacks - NumCrits)!])
In the case of 30/100 crits with a 30% crit chance, this is:
.3^30 * .7^70 * 100! / (30! * 70!) = 8.68%
And this tells us... What? Pretty much nothing. The odds of getting EXACTLY the right number of crits is relatively low. Hopefully you already knew this. But see, we're looking for something more. We want the chance of a range of values. For example, the chance of getting AT LEAST 10% more damage means the chance of getting 43 to 100 crits. Ultimately, we have to use summation. As in, we have to pick a total and a crit chance, then calculate it for each integer value. I don't like doing that by hand, so I wrote a small program to do it for me. I won't be reproducing intermediate steps, just end results.
43-100% crit, or 10% or higher improvement: .4% chance
0-17% crit, or 10% or higher drop: .2%
37-100% crit, or 5%+ up: 8% chance
0-23% crit, or 5%+ down: 7.5% chance
Hmm, with 100 casts, the odds of a 10% increase are almost nil and the odds of a 5% increase are pretty unreasonable. But nothing right now takes that long. For example, with a 2 second cast spell, killing Patchwerk in 2:30, we're looking at 75 casts. And the burst phases of most fights are much shorter, for example the Tenebron phase of +3 should be lasting less than 45 seconds or MAYBE 30 casts. So lets say 40% crit, 30 casts. Midpoint should be 12, 16 is roughly a 10% increase.
30 casts, 40% crit
16-30 crits: 9.7% chance
0-8 crits: 9.4% chance
14-30 crits: 28% chance
0-10 crits: 29% chance.
Interesting. Even on a short fight with a high crit chance, the odds that you'll crit significantly more than you should is pretty low; a 10% increase is definitely possible for an individual, but you'll almost never see the whole raid get even a 5% DPS increase.
I'm quite surprised, I was expecting to see crit give a much higher variance. This is actually pretty cool, since it means that a bad group will almost never get lucky and do a ton of DPS, and a good group will almost never have terrible luck and lose a ton of DPS. Which means if you're seeing variance in your raid's DPS, you should probably look at external factors, like void zones spawning three deep around your mobs.
Edit: Source for the program is here-