So I'm realizing I'm bad at game theory as I'm trying to figure out how frequently Hawkmoon is better than any other hand cannon in crucible. 2 headshots to kill is extremely strong and you want to maximize your chances of landing it. I'll walk through what I have and someone let me know if I messed it up: From a full clip your odds of NOT getting a bonus bullet are 10 in 13. Your odds of not getting a bonus bullet a second time are 9 in 12. 10/13 * 9/12 = .577 1 - .577 = .423 or a 42% chance of killing someone in 2 headshots Assuming all shots are on target, the one place you don't want to see a bonus bullet is in your third shot because it's effectively wasted and you'll have to reload either way because you're left with only a 38% chance of proccing the bonus bullet at an opportune moment instead of 42%. If you go through three shots and don't proc the bonus bullet, then obviously you don't want to reload because you're looking at 47% odds of a 2 shot kill in your next encounter. If you miss one shot from a full clip and don't see a bonus bullet I'm going to guess it depends on when you missed? If I miss bullet 1 and don't see a bonus bullet (BB from here on out) then I have a 3/13 chance of having missed a BB shot and now have 9 bullets in my clip. (3/13)*(7/9)*(6/8) = 13% chance I do not see a BB at an opportune time and have 2 left. 10/13* 6/9 * 5/8 = 32% chance I do not see a BB at an opportune time and have 3 left Leaving me with a 55% chance of proccing a BB on my second fight if I don't see one in my first. Despite potentially unknowingly wasting a BB my odds of seeing a BB are still higher in fight 2. That's interesting so let's check out the numbers in fight 1: 42% chance of not proccing and I have 3 BBs left. 15% chance of not proccing and I have 2 BBs left. My odds of seeing a BB *at all* go up by firing my first bullet into the ground, only slightly for my first encounter but then higher for my second. I'm bad at this and am supposed to be studying physics right now, so I'm going to stop for now but am I doing this right? EDIT: TL;DR for people who don't realize I'm trying to help them: Missing on purpose and reloading at certain times will increase the odds of finding lucky bullets and I'm trying to find those points for you.