Say we have many bad things that can happen, and we would like all of them not to happen. For each thing we know the chance that it happens. How can we show that we can avoid every bad thing?

One way is if all the chances are small, and there are not too many bad things. In the worst case, every bad thing happens alone, so that two bad things never happen together. Then to find the chance that a bad thing happens, we could add up all the chances for all the bad things. If this is less than one, then there must be a way for no bad thing to happen.

We can sometimes do better if most of the bad things do not care about each other: if the chance that one bad thing happens only changes because of a few other bad things (one way this can happen is if all the bad things are about dice rolls, and there are only a few bad things about each die). Now if we only add up the chances for the bad things that one bad thing cares about, and this is still small (less than one in four), then we can still avoid all the bad things, no matter how many there are!