The question relates to statistics, and how you add up odds.
I'll spare you my function for determining the odds of a die roll. Assume for the sake of simplicity that the foe's AC is 11, and my attack mod is +1, which would mean that a 10 or higher will hit. In this case, the odds of a hit are 0.55
Let's say my first attack is with a dagger; The average damage of a d4 will be 2.5 plus my mod of one. The odds to crit are .05, so I'll subtract that from the odds to hit, calculate it separately, and add the results together. This leads to:
A1: 0.5 * (2.5 + 1) + .05 * (2 * 2.5 + 1) = 1.75 + 0.30 = 2.05
This means my average damage for attack A1 will be just over 2.
My second attack is also with a dagger. It has the same chance to hit, but I don't get the +1 modifier on the damage roll:
A2: 0.5 * 2.5 + .05 * (2*2.5) = 1.25 + .25 = 1.50
This means my average damage for attack A2 will be 1.50
Add these together and I get my average damage per turn of 3.55. Now, ignoring advantage and disadvantage, which I've already figured out how to calculate, this would be my total. However, where I'm having trouble is factoring in my odds to get a Sneak Attack. For the purpose of this experiment, I am positioning myself well and can potentially get a sneak attack on hit every single turn, from either A1 or A2 (Yes; Two handed fighting allows the second strike even if the first misses). The odds of either attack A1 or A2 hitting are .7975 or 79.75% (.55 + .55 – .3025, which is the odds of both attacks hitting). Thus, the average damage of a sneak attack at level one (1d6) is:
S1: .7975 * 3.5 = 2.79125
This means that on average, every turn I will deal 2.79125 in Sneak Attack damage.
My first instinct is to add this figure directly to my figures for A1 and A2, meaning that my average total damage per turn will be 6.34125.
However, And here's where my mind is starting to boggle, A1 and A2 are independent events. Either or neither can hit. S1, the Sneak Attack is…sorta independent??? I guess? Maybe? I'm not sure if I'm properly adding the odds of S1 to the odds of A1 and A2.
Thoughts?
Best Answer
The bad news: The variables are dependent. If you miss with both attacks, you're not getting any Sneak Attack damage.
The good news: If you're only interested in the average damage (specifically mean damage, aka expected damage) as opposed to e.g. the entire probability distribution, linearity of expectation (reference 1, reference 2) applies even if the variables are dependent. So you can in fact just add up the expected damages of the first attack, the second attack, and the sneak attack to get the expected total damage.