For those of you who might be wondering about the worth of the Assault skill Rapid Fire, here's the skinny on how the math breaks down.

Rapid Fire has a 15% penalty to Aim, so, for a given 75% normal shot, Rapid Fire will fire two 60% shots.

Assuming that your shot will deal 1 damage, for simplicity's sake, if you fire your gun 100 turns, on average you'd deal 75 damage with a normal shot and 120 damage with Rapid Fire.

This is because with Rapid Fire you have a 36% chance of hitting twice (0.6 * 0.6), a 48% chance of one shot hitting and one shot missing (0.6 * 0.4 + 0.4 * 0.6), and a 16% of both missing (0.4 * 0.4). The damage output there over 100 turns is 2 * 36 + 1 * 48 = 120.

So, on average, you want to be Rapid Fire-ing for most of your shots to increase your damage.

Just don't forget to reload twice as often.

EDIT: As a corollary, in this example, since your chance to miss both shots is 16%, then you want to be Rapid Firing in higher leverage situations too since your chance to hit at least once is 100 - 16 = 84%

Oh god I hate math

@Sackmanjones:tl;dr version: Always Be Rapid Firing

That's not how the combat system works though. It doesn't roll a die for each attack.

@Dezztroy: The RNG might seed at the start, but each action advances that seed. Firing twice in Rapid Fire means two actions which means two random numbers to check against.

Whenever I really need something to die or when I'm pretty close. It sounds like it would never hit but I use it all the time and it can really save your ass, keep an eye on your ammo though.

@Dezztroy: Not to mention that even if you're right and they don't know how to code their game, then it's still 120 vs. 75 because those two hits will both definitely happen 60% of the time. They're independent events either way.

I love the math behind rapid fire. I've been doing P(A U B) = P(A)+P(B)-P(A ∩ B) in my head to see which is better if I only need 1 hit to kill.

You could always plan out attacks in conjunction with a Heavy + Holo Targeting, but then at that point you might not even need to try to rapid fire.. but maybe it's something hefty like an elite or something with an innately high dodge chance.

@Laivasse said:

@ElCapitan: Yep, I think the maths of rapid fire are worth pointing out, in that there's hardly ever a reason not Rapid Fire other than ammo conservation. The point where the probability of missing both Rapid Fire shots starts to outweigh the probability of missing one non-RF (+15%) shot, doesn't happen until the Rapid Fire percentage is as low as around 18%. Chance of missing both in those circumstances: 0.82*0.82=0.6724 or 67.24%. The non-RF shot in that situation would be 33%, which means 67% chance of no damage. I think that's the threshold at which RF becomes more likely to do no damage at all. The chance of missing both at 19% is 65.61% vs 66% for the non-RF shot.

I usually stop using rapid fire below around 35% chance to hit (50% non-RF), simply because the cost in wasted ammunition starts to outweigh the marginal extra hit chance, in my opinion, especially if I'm already low on ammo. That's probably less important after you've researched the ammo upgrade at the foundry, or if you think you'll have an opportunity to reload soon though. Of course, good positioning often means your chance to hit is closer to 85% without RF, so I'm almost always using it anyway.

@Laivasse said:

@ElCapitan: Yep, I think the maths of rapid fire are worth pointing out, in that there's hardly ever a reason not Rapid Fire other than ammo conservation. The point where the probability of missing both Rapid Fire shots starts to outweigh the probability of missing one non-RF (+15%) shot, doesn't happen until the Rapid Fire percentage is as low as around 18%. Chance of missing both in those circumstances: 0.82*0.82=0.6724 or 67.24%. The non-RF shot in that situation would be 33%, which means 67% chance of no damage. I think that's the threshold at which RF becomes more likely to do no damage at all. The chance of missing both at 19% is 65.61% vs 66% for the non-RF shot.

I usually stop using rapid fire below around 35% chance to hit (50% non-RF), simply because the cost in wasted ammunition starts to outweigh the marginal extra hit chance, in my opinion, especially if I'm already low on ammo. That's probably less important after you've researched the ammo upgrade at the foundry, or if you think you'll have an opportunity to reload soon though. Of course, good positioning often means your chance to hit is closer to 85% without RF, so I'm almost always using it anyway.

@ElCapitan: So would a generalized form of that be average damage of a rapid fire shot is equal, when A(X) is average damage in terms of the chance to hit X, to A(X) =2(X - 0.15)^2 *the odds of hitting twice* + 2(X - 0.15)(1.15 - X) *the odds of hitting once*? That comes out to A(X) = 2X - 0.3 as the average damage for a rapid fire shot. This is assuming each shot does one damage, although it should hold true no matter what you pick for per shot damage.

If we turn the equation in to the difference between the average damage of RF vs normal fire, we get 2X - .3 - X, or X - 0.3. This indicates that if the chance to hit is .3 (or 30%), the average damage is the same, making RF the worse choice (greater ammo use). For any value to hit less than 30% RF is definitely the worse choice, with lower average damage and more ammo use.

This only really matters when you are dealing with a situation where you are trying to maximize damage. If you only need one hit (e.g. just a couple of health pips on your target), you use 1-(1-(x-.15))^2 - X, which says that as long as your chance to hit normally is between 33.4% and 96.6% (approx), which is basically what @Laivasse said, it is better to use rapid fire. Note there is a higher end bound there, which there isn't if you are just going for max average damage.

It's been a while since algebra so someone let me know if my math is off.

@PopBot: You're right, thanks. I tend to ignore lower probability shots since, well, you can probably put yourself in a better situation than taking that shot.

In my experience, taking such a low probability shot on Classic difficulty leaves you in a vulnerable position (a lot of aliens, especially with Blood Pact or psionics) can still wreck your day even if you've only got a < 30% chance to hit. Probably better to retreat/overwatch or hunker down, depending on plenty of other variables.

@ElCapitan: Definitely, this is all pure math. If I was sitting at a 35% probability shot, or worrying about ammo, I wouldn't be breaking out my tables, I be running for cover!

*edit: also this doesn't factor crits in, which might skew the average damage equation towards using RF more (I can't remember if RF alters crit chances).

Wow. Well thanks, smart dudes, for working this out for me. Tips appreciated!