normzone, this is for you. I hope it's what you were looking for.

Instead of talking about system reliability and probability, I'll use an example that will hopefully be easy to digest and then do a "bait and switch" at the end.

Let's say you and your partner like to play tennis doubles Saturdays at 10 AM. You know two people, Bob and Jane, and Bob and Jane are not a couple, but they both play tennis and can play together as a double.

The chances that Bob will show up to the club are 50-50 and for Jane the same, 50-50.

What are the chances that you'll get to play against Bob and Jane? Should be pretty easy to see that it's 1 in 4, or 25 %, right? Because they could both be no-shows, Bob could show up but not Jane, or Jane but not Bob, or both of them. Those are the only four possibilities, and only one of those four means you get to play.

Now let's say that one of them is 100 % reliable, and make it Bob. Now it should be easy to see that the chances of having them to play against is 50 %, because it's only a matter of whether Jane shows up, which is 50-50.

If the chances of Bob or Jane showing up depend on the time of day, say zero at 6 AM, 50-50 at 10 AM, 75 % at 2 PM etc, you can have time-dependent reliability information which you use to make the calculation. If you can find an equation which gives R as a function of t, you can use that R(t).

So far so good?

Here comes the switch. Instead of asking whether two people are reliable enough to show up at the same time to play tennis, make it a more realistic system. Say you have a machine that makes donuts. There are three steps, extruding the dough into a donut shape into the hot oil, keeping the oil at the right temperature, and lifting the donuts out of the hot oil at the right time. If any of these three steps isn't done right, no donuts. This is an example of a system with three subsystems in series. Just like the tennis players, each of the three subsystems has its own probability, and if you play with simple probabilities (like 100 % or 50-50 for each of the three subsystems and write out the possible outcomes, you should be able to convince yourself, or someone who isn't good at math, that you multiply the individual probabilities to get the system reliability.

Hope that hit the mark.