This is for all you non-mathletes out there who want to be able to perform some basic calculations. I find it helpful to use Excel when I start to get into trickier calculations - it's amazing what you can do with a spreadsheet.
WARNING: This is going to start out really basic. Don't eye-roll, some people don't know how to do this stuff. We will get to slightly trickier stuff in later parts but if you are any kind of mathlete this stuff might be review.
If you ARE a mathlete, however, feel free to check my work! ;D
Anyways, let's start with something simple. We have a BS 4 bolt pistol firing a single shot at a Space Marine. We want to know what the probability is that it will kill the space marine.
You can think of each of the steps here as a single "gate." Each "gate" is narrower or wider depending upon how difficult it is to get through. We are trying to see how likely it is to get through all the gates to the end of the path. Ok?
First step is to calculate the "Hitting Gate." So at BS 4 we hit on 3's, 4's, 5's and 6's. We miss on 1's and 2's. That means that we will hit FOUR out of SIX times, or two thirds. So we take our one shot and multiply it by two thirds and get... two thirds.
Second step is the "Wounding Gate." A bolt pistol wounds on 4's, 5's and 6's. So we wound three out of six times, or one half. We take the two thirds that we got from step one and MULTIPLY it by the one half from step two... the result is one third.
Step three is the "Save Gate." What we are trying to find is successful wounds (we are the ones firing the bolt pistol). So what we are looking for is the rate at which our bolt pistol will get through the Marine's save. The Marine will fail his armor save on a 1 or a 2, so two out of six times, or one third. We multiply the one third from step two by one third and get one ninth (1/9 or about 11%).
Written out that's (2/3) * (1/2) * (1/3) = 1/9
So one out of every nine shots from a bolt pistol, on average, will kill a Space Marine.
Once we have that basic ratio, we can multiply it by the number of shots we are firing to get an "average" number of unsaved wounds caused for that many shots. So if we fire nine shots, we will "average" one unsaved wound. A tactical squad firing 20 shots from rapid-firing bolters will "average" 2.22 unsaved wounds against Marine targets.
Ok, that's pretty simple, right?
Well, how many shots does it take before we can "guarantee" we will cause one unsaved wound?
Many people will tell you nine. They are wrong. There is no number of shots where we can guarantee that we will cause one wound. We can fire one shot and get one wound, or none. We can fire 40 shots and cause 40 wounds... or none. All possible outcomes. The "average" number of wounds is helpful to know, but don't take it as gospel.
We can, if you wish, determine what the odds are of causing "at least" one of something, though. That can be VERY important when dealing with, say, vehicle effects.
Part two we will deal with the other side of this question - how likely is it that if you fire multiple shots you will get the result you are looking for.
UPDATE: Part two can be found here.