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.

-Brian