We all know the situation. A Sectoid is being eyed up by a high-ranking Assault with a fully-loaded shotgun. He shoots once... and misses! But what if he had shot twice, with the 15% penalty to aim to each shot? Would it have made the odds of hitting more likely, or less?

Below I've outlined a mathematical method of figuring out when (in terms of raw probability) you should use rapid fire or not.

Rapid fire carries a 15% penalty to aim; however, the assault gets to shoot twice. We want to see when this carries a statistical advantage.

Let the odds of hitting with a regular shot = x. Therefore, the odds of hitting with one rapid fire shot = x - 0.15
(I used 0.15 instead of 15 as it makes the maths a lot easier to deal with. We can multiply by 100 at the end to give the percentage.)

We want the assault to hit at least once, or to put it another way, not miss both shots. The odds missing both shots are calculated by
(1 - (x - 0.15))²
= (1.15 - x)²
= 1.3225 - 2.3x + x²

Therefore, the odds of hitting with at least one of those shots are
1 - (1.3225 - 2.3x + x²)
= -0.3225 + 2.3x - x²

We want to find when this is greater than taking a single shot, so...
-0.3225 + 2.3x - x² > x
-0.3225 + 1.3x - x² > 0

Using the quadratic formula, "x = (-b ± √(b² - 4ac)) / 2a" and a = -1, b = 1.3, and c = -0.3225, we get that 0.333 < x < 0.966.

If you want to maximise your chances of hitting something, use rapid fire when the odds of hitting something with a normal shot are between 97% and 33% (exclusive). It's up to you to apply that in terms of ammo consumption, enemies using covering fire, and the sheer unhappiness and disbelief you may experience should both shots miss (which is at most 67% when sticking to my rules).

Good luck, Commander.