How can I calculate this expected rate?

Question

In DotA, there is a character called "Axe". Every time he is attacked, he has a chance to spin (17%) his blade and deal damage based on what level the skill is, 100 / 125 / 150 / 175 damage for levels 1, 2, 3, and 4. When the spin activates, it triggers the cooldown of 0.7 / 0.65 / 0.6 / 0.55 seconds, so that attacking Axe does not generate a chance to spin.

I was trying to calculate an average damage per second that this skill generates, given that most of the time you will find Axe taking around 5 attacks per second (average 1.667 attacks per second from creeps, 3 creeps per camp), so I figured that would mean the probability of him spinning in any second is 1 - 0.83⁵, so given the damage from earlier, we should expect him to deal 60.61 / 75.76 / 90.91 / 106.07 damage per second?

I got this calculation, but then I realized that I have to factor in the cooldown somewhere, but I have no idea where to start.

Answer 1

Let r be the chance an attack caused axe to spin and d be the damage it does. Since we know the number of attacks on Axe per second, we just need to find the expected spin damage per attack, call this x. If someone attacks Axe once, we expect the retaliation damage to be rd. However, each attack blocks a certain number of attacks depending on cooldown, call this number b. This reduces our expected value by brx. So we have:

x=rd-brx
x(1+br)=rd
x=rd/(1+br)

Note that this assumes the combat goes for an infinite amount of attacks. When the combat is shorter, the expected damage will be higher. Given the speed that combat happens in Dota, this effect will often be significant