Rising probability question
Before I get to my question let me give some context. I am playing a video game where critical hits score much more damage than regular non critical strike hits that just deal basic weapon damage. There are two stats that I am concerned with that I can get on my weapons attacks speed and critical hit damage. I have a base attack speed of 2 attacks per second, and have a base crit damage of 235 percent weapon damage, and a critical strike chance of 3%. So with 100 damage on my weapon I will do 100 damage when I do not crit which will happen 97% of the time, and I will do 235 damage when I do critically strike which will happen 3% of the time. My critical hit damage and attack speed are always changing as I get new gear. I am trying to figure out whether or not I want to sacrifice some crit damage for attack speed or just stack crit damage and use a passive skill that raises my crit chance over time until I land a critical hit, at which point my crit chance will be reduced back to its original 3 percent. So there is a little context. Thanks for reading this far. When I can answer the question below I will get a better idea of how I will want to proceed in gearing my character.
HERE IS THE QUESTION
How many seconds (on average) will it take to land a critical hit when:
a.) The 1st second consists of two attacks each of which have a three percent chance to critically hit, and
b.) The 2nd second consists of two attacks each of which have a six percent chance to critically hit, and
c.) So on and so forth with the the critical strike chance raising three percent per second until a critical strike is landed?
So, the third second would consist of two attacks each of which had a 9 percent chance to critically strike and the 4th second would consist of two attacks (the 7th and 8th attacks) each of which had a 12 percent chance to critically strike. I think you get the point.
My thoughts are that on average it would take 16 and 2/3 seconds to land a critical strike since that is how long it would take for the critical hit chance to reach 50 percent assuming I went that long without landing a critical hit. Since sometimes I would crit before and other times I would crit after I made it to the 50 percent critical hit chance mark. This seems too easy of an explanation however, especially considering the fact I get two attacks each second and on the 16th second I would attack twice when each of these two attacks would have a 48 percent chance to crit. Part of me wants to think that it would be more likely to crit by the 14th second than to not crit by the 14th second since I would have to miss 6 times in the 12th 13th and 14th seconds where the probability was 36%, 39%, and 42% respectively. It seems that with even a crit chance of 33.3 percent during these three seconds coupled with the fact I would have 6 chances the odds would be in my favor to crit during this time.
What am I missing here. I would love to see some sort of formula, although I prefer to have the reasoning behind a problem such as this be explained to me in words.
Thanks so much guys hope this at least stimulated at least a couple brain cells a bit since you took the time to read this.
Re: Rising probability question
I already discussed this topic in a german board. I do not think that there is a nice closed form for the probability of having x seconds to the next critical hit (related to the Pochhammer symbol, which is just a product of all the chances to have no critical hit).
However, we did monte carlo simulations to express the bonus from sharpshooter (which lets the chance to get critical hits rise) as bonus to the base chance of critical hits.
A graphical representation is here, but I think this still contains a bug we fixed later. As table, rows are attacks per second, columns are different base chance for critical hits and the entries are absolute values:
|aps ||5% ||10% ||15% ||20% ||25% ||30% |
|4 ||0.0503 ||0.0374 ||0.0295 ||0.0234 ||0.0200 ||0.0170 |
|3.8 ||0.0487 ||0.0373 ||0.0282 ||0.0211 ||0.0167 ||0.0137 |
|3.6 ||0.0531 ||0.0399 ||0.0309 ||0.0234 ||0.0189 ||0.0152 |
|3.4 ||0.0593 ||0.0440 ||0.0354 ||0.0294 ||0.0234 ||0.0189 |
|3.2 ||0.0633 ||0.0456 ||0.0382 ||0.0320 ||0.0258 ||0.0212 |
|3 ||0.0586 ||0.0471 ||0.0369 ||0.0309 ||0.0243 ||0.0206 |
|2.8 ||0.061 ||0.045 ||0.0367 ||0.0288 ||0.0226 ||0.0176 |
|2.6 ||0.0654 ||0.0506 ||0.0394 ||0.0334 ||0.0267 ||0.0206 |
|2.4 ||0.0717 ||0.0569 ||0.0452 ||0.0392 ||0.0305 ||0.0282 |
|2.2 ||0.0803 ||0.0624 ||0.0523 ||0.0423 ||0.0359 ||0.0320 |
|2 ||0.0737 ||0.0605 ||0.0507 ||0.0419 ||0.0363 ||0.0306 |
|1.8 ||0.0776 ||0.0607 ||0.0504 ||0.0393 ||0.0321 ||0.0266 |
|1.6 ||0.0863 ||0.0688 ||0.0577 ||0.0462 ||0.0398 ||0.0332 |
|1.4 ||0.0987 ||0.0847 ||0.0682 ||0.0606 ||0.0513 ||0.0449 |
|1.2 ||0.1107 ||0.0930 ||0.0812 ||0.0727 ||0.0626 ||0.0558 |
|1 ||0.1075 ||0.0940 ||0.0809 ||0.0684 ||0.0589 ||0.0546 |
|0.8 ||0.1159 ||0.1013 ||0.0864 ||0.0732 ||0.0618 ||0.0542 |
|0.6 ||0.1434 ||0.1268 ||0.1090 ||0.1000 ||0.0859 ||0.0738 |
Example: With 2 attacks per second and 5% base crit chance (which is the minimum), sharpshooter gives you an effective bonus of ~7,4% critchance.
However, this is just true if you shoot with single attacks and without any break. Attacks which hit multiple targets are similar to a higher attack speed, and running around from time to time increases the bonus.
Usually, you will have a critical hit much earlier. Simple example: 10 shots with ~10% chance already give a ~70% chance to have a critical hit somewhere. With 2 attacks per second, you get these 10 in the range of ~7.5-12.5%, far away from 50%.
My thoughts are that on average it would take 16 and 2/3 seconds to land a critical strike since that is how long it would take for the critical hit chance to reach 50 percent assuming I went that long without landing a critical hit.
I still have my simulation code, so I can produce more values or other related data if you like.
Re: Rising probability question
Thanks this clears a lot of this up. Can you show your calculation proving your 70 percent calculation in your next to last paragraph? Then I will be 100 percent satisfied. :P. Thanks again.
Re: Rising probability question
The chance to not have a critical hit is 90% each shot. Therefore, the chance that all 10 shots are not critical is 0.9^10 = 0.349 and the chance that at least one shot is critical is 1-0.349=0.651. While this is a bit below 70%, 11,3% chance to have critical hits would give 69,9%.