Sounds interesting, and my google fu fails in finding me a internet explanation that goes into more detail (what I'm finding simply seems to be finding a "at most" function, finding the odds of the outcome being less than or equal to a certain X, which doesn't assist in finding probabilities that are dependent on prior probabilities). I would like an example of you applying the process to stuff from this quest, if you wouldn't mind.
Dunno about logarithms but I can demonstrate the algebra.
In order to roll an X crit you must first roll an X-1 sized crit.
This means that we can say that the outcomes are dependent as DrTortoise said.
You calculate the probability of dependent events by multiplying the probability of the events as if they where unrelated, so for example, two heads on a coin is .5*.5=.25.
When we had a +40 training modifier we passed that on any roll from 60-100 so we had a 41% chance of criting, which is the independent chances of a roll.
In order to have a 11X crit we need to roll a +100 11 times in a row. So the probability of that multicrit was (41/100)^11 which give us 0.006%.
Technically this is calculating the chance of getting an 11 crit or higher. If we want to know the odds of 11 specifically we would then multiply by the chance that we don't get another crit roll after this.
This totally ignores crit overflows, because no matter what DrTortoise said, there is not an easy way to calculate the probability of dependent events when the probability of the events is dependent on prior rolls.
However when thats the problem, you write a computer script to simulate it, brb.
Code:
#Simulates rolls for Uchiha Quest: Silver Edition
import random
bonus=40
sampleSize=500000
def overflow(excess):
roll=random.randint(0, 100)+bonus+excess #roll d100
if roll>100:#if roll is greater than 100, calculate excess and roll crit die
excess=roll-100
crits=overflow(excess)
return roll+crits
else: return roll
targetDC=1100
hits=0
for i in range(sampleSize):
if overflow(0)>targetDC:
hits+=1
print(hits/sampleSize*100)
My python is kinda shit but heres what I've got. I assume that when DrTortoise talks about logarithms he's done something similar to this at a lot of data points and calculated a best fitting line.
This calculator gave me a probability of 5.643% so that's orders of magnitude bigger than the chance without overflow.
Postscript: While testing this script I hit the maximum recursion depth consistently within the first 10 to 40 rolls of a +50 bonus. I don't know how DrTortoise intends to handle this but in the future I think that bonuses we are given should probably only be valid to a certain depth. +50 in particular is kind of an important milestone because at this point the mean roll on a d100+the bonus is itself a crit can be reliably self sustaining without overflow, thus allowing overflow to build to ridiculous amounts. Yeah I can see crits in the range of 4850
[X] Your training with your father is going quite well, why stop now?
Train with Uchiha Ryota. Train 'Taijutsu' and Uchiha Interceptor Fist' with an additional +10 to each roll.
[X] Not only is it useful, it is also extremely enjoyable. Moments like these will become rarer with age. Make the most of the ones you have, especially if they simultaneously help you improve your skills.
Train with Nakamura Shiori. Train 'Chakra' and 'Mind's Eye of Kagura' with an additional +10 to each roll.
[X] You had a lot of fun in the Forest of Death with your mother, see if she wants to go again?
Train with Nakamura Shiori. Train 'Perception' and 'Tracking' with an additional +10 to each roll.
[X] From what you understand the Uchiha have a 'rite of passage' involving Katon ninjutsu. This technique is available to anyone in the clan, regardless of age. What better time to learn it than now?
Train 'Ninjutsu,' 'Katon,' and 'Chakra Control.' 5% chance to unlock skill 'Katon: Gōkakyū no Jutsu (Fire Release: Grand Fireball Technique)'
[X] Your father is officially the only Uchiha you like. Spending time with you should be a proper reward for his accomplishment.
[X] You didn't have such a great time the last time you tried to interact with children your own age. Maybe take it a bit more slowly this time? There's supposed to be another genius child around your age, you should see about getting to know him.
Sure because I'm sure Kakashi will be less insufferable than he was while young in canon.
Seriously though after the last fail how could I turn down another.