[PvK]

Multiplication is the way probability actually works for independent factors in mathematics and the universe at large.

For example, I remember asking a kid in high school if he needed a ride home. His understanding of probability was like Space Empires', as he demonstrated by explaining, "No. My mom has a 50% chance of giving me a ride, and John's mom has a 50% chance of giving me a ride, so I have a total 100% chance of getting a ride home." It was actually hard to convince him that there was any chance neither would give him a ride home (perhaps though because he realized he was being silly and making a bad argument to get my goat, but that's the way SE4 works, which does get my goat and decrease my interest in it).

Assuming he was right about the 50% chance of each (a silliness in itself), the total chance was actually 75%, because the factors don't add to each other. One has a 50% chance of coming through, and the other only matters if the other fails, so its 50% only applies to the remaining 50%.

The math is:
Total Chance = 1.0 - (1.0 - Chance A) x (1.0 - Chance B)

That's for independent positive chances of causing something. Things like, if your computer has a 60% chance of finding out how to fix the Hyperdrive, and your Chief Engineer has a 30% chance of finding a solution on his own, you'd get:

Chance = 1.0 - (1.0 - 0.6) x (1.0 - 0.3)
Chance = 72%

This passes the following sanity checks:
* The total chance is less than 100%, since neither was certain.
* The chance is higher than either independent chance, since both provide a chance if the other fails.
* Note that if you think of 100 possible outcomes, 60 of which are covered by the computer, that leaves 40 not covered by the computer. 30% of _those_ will on average be covered by the engineer, so .3 x 40 = 12, and 60 + 12 = 72. Perfect.

Now, for negative (preventative) causes. Suppose your weapon targets accurately 70% of the time, but my shield deflects 90% of all hits (so only 10% get through). In that case, of 100 expected average tries, 70 will be potential hits. The shield applies in _those_ cases, because it's a negative effect dependent on a hit. Of 70 well-targeted shots, 90% get deflected, 10% get through, so 7 on average will be accurate AND get through.

You could do this with two random number checks, or you could just multiply:
(chance of good aim) x (chance of avoiding deflection)

chance of success = 0.7
chance of failure = 0.1

Just like for multiple things that lead to a success, there can be multiple things that can lead to a failure. Maybe you have an ace gunner who corrects computer targeting errors 80% of the time, but the enemy also has ECM that jams your view and prevents a hit 10% of the time. Assuming these are all accurate statements, the math for this is:

chance of a hit = causative chance x (1 - preventative chance)

where:
causative chance = 1.0 - (1.0 - computer targeting) x (1.0 - gunner correction)
and
preventative chance = 1.0 - (1.0 - shield deflection) x (1.0 - ECM jamming)

I get:
chance of a hit = 0.94 (chance the computer or gunner get an accurate shot) x (1 - 0.91) = 8.46% chance of a hit which gets past the deflector.

Note in this kind of system, the 90% deflector shield is obviously a very powerful effect that as you would expect, dominates the chances. But, also as you would expect, it will NEVER make it entirely impossible to get through - the chance will just always be under 10% as long as that deflector is having a 90% chance of blocking. As long as all the causative factors are greater than 0.0, and all the preventative factors are less than 1.0, there is always a chance of something happening or not. And, that chance makes sense, assuming your values are correct.