Critical Strikes, Timed Procs, and YOU

Basic <The Collective> of Kil'Jaeden


So your critical strike chance... you just multiply the number of hits by your chance and that's it right?

Well not quite... say you have a proc on your crit and that proc lasts 15 seconds. What crit rate do you need to maintain the proc? 15 seconds, 10 casts, does that mean you only need a 10% crit rate to sustain it? No, more likely than not you will only keep your proc up 2/3 of the time.



The solution lies in the probabilities. First your chance to get at least one crit over time is the formula F(crit,swings) = 1 - (1 - crit) ^ swings. So if you swing 10 times and have a 10% crit there is a 65.1% chance to get one crit in those 10 swings. It works this way because your chance to get just one crit is the opposite of your chance to miss all of them, so your chance to not get a crit once is one minus your crit chance. Multiply that by the number of swings and that's your chance to not land a single crit. Now subtract that number from one and you have the chance that at least one attack will be a critical hit.

Now I clearly have forgotten all my calculus, and tho I got close my TI-92 correctly tells me that swings needed = F(crit,chance) = ln(1 - chance) / ln(1 - crit).

If you want to figure your interval of a 50% chance then the formula gives you these percentages for the number of swings needed given your crit rate: 11% for ~6 casts , 13% for ~5 casts, 16% for ~4 casts, 20.5% for ~3 casts

Now that gives you a start, but for your procs to really stick you want 75% or 90% chance. So for 90% you get: 14% ~15 casts, 16% ~13 casts, 20.5% ~10 casts, 25% ~8 casts, 28% ~7 casts, 32% ~6 casts, 37% ~5 casts, 44% ~4 casts, 54% ~3 casts

You can plot it out for your current stats fairly easily. The formula you want to determine your target crit rate based on number of swings is F(swings,chance) = 1 - (1 - chance) ^ (1/swings)


For example: Nightfall procs 4% of the time on corruption or drain life. With both churning you get 4 attempts every 3 seconds. At 13 seconds you have around a 50% chance of having seen your nightfall proc!

For me, I'm watching inspiration... so I want to beat 90% in 15 seconds, or 10 casts. With talents and gear I want to be pushing 20.56% chance to crit to be able to reliably keep my proc up on the main tank.

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FORMULA SUMMARY

I know my crit rate, how often will I crit?
This is how often you have a 50% chance to land a crit:
F(crit) = ln(0.5) / ln(1 - crit)

I know how long my proc lasts, what crit rate do I need?
figure out the duration of proc by the number of swings/casts you can make over it's lifetime...
then this is the number of swings needed to have a 90% chance to refresh the proc:
F(swings) = 1 - (.1) ^ (1/swings)

I can get so many swings in pretty quick, what is my chance to land a crit off one of them?
F(swings,crit) = 1 - (1 - crit) ^ swings

I have three swings that have a better chance to crit then some extra swings I know I'll get in. How would that work?
F() = 1 - (1 - crit(swing1)) * (1 - crit(swing2)) * (1 - crit(swing3)) * (1 - crit(normal))^extra_swings

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by the way warlocks... this is why pyroclasm works as it does where the chance is spread out over the durration of the effects:

rain of fire, 4 "swings" - 7.25% chance to proc per hit (nice round number there)
hellfire, 15 "swings" - 1.987% chance to proc per hit (close enough they probably use 2%)


If you looked at nightfall the same way, drain life would have an 18.46% chance and corruption would have a 15.07% chance. 26% ain't too bad in comparison.