How do you calculate the mean value for a percentage chance? For example: Damage per second with an item that has crit chance?

Question

To elaborate on the example: if I have a weapon that does 6 damage, and attacks 1 time per second the calculation is damage * attacks per second. But what if the weapon has a 5% chance to strike critically for 150% damage? Now the expected damage goes from 6/second to something like 6.1 per second. What equation can I use to find the expected damage?

Answer 1

You get with your example

$$(1.50 \times 6) \times 0.05 \times 1 + 6 \times 0.95 \times 1 = 6.15 \tag{1}\label{eq1A}$$

damage per second. This is due to it having a $5\% = 0.05$ chance for the $150\% = 1.5 \times 6$ probability of damage, with the only other possibility being a $100\% - 5\% = 95\% = 0.95$ probability for the regular damage of $6$. In general, to get the expected damage per second, you sum the products for each possibility of its damage, the probability of it occurring, and the number of attacks per second.