The 'sum rule' is what you use when the probability of one event precludes the possibility of the other events from happening. This is usually the rule that we should use when we are asked to calculate the probability of 'either x or y'.
In case this sounds like Greek to you, let's consider a simple example. Say you are playing a game show in which there are three doors, behind one of which is a prize. Assume you have no idea which door holds the prize. What are the odds of the prize being behind either door number 1 or door number 2? (Note "either ... or".)
Since the odds of the prize being behind any one door must be 1 in 3 (0.33), the odds of it being behind any two of the doors is:
0.33 + 0.33 = 0.67 (rounded off)
Let's consider a second, slightly more interesting example:
A ball is drawn at random from a box and contains 10 red, 30 blue, 20 white and 15 orange balls. Find the probability that it is red or white or blue.
First we calculate the odds of it being red. There are 75 balls in total, so the odds of me randomly picking out a red ball must be 10 out of 75 (10 ÷ 75 = 0.13). Similarly, the odds of be getting a white ball must be 0.27, while the probability of me finding an blue ball is 0.4.
Putting this together, we can use the 'sum rule' to work out that the answer to the question is:
0.13 + 0.27 + 0.4 = 0.8 (i.e. 80%)
(Incidentally, the more mathematically-inclined people may have thought of a clever short cut. The odds of finding an orange ball are 15 ÷ 75 = 0.2. Therefore the odds of not finding a blue ball must be 1 - 0.2 = 0.8. This is the same answer, of course, since the odds of not finding a blue ball is simply another way of saying the odds of finding a red or a white or a blue ball. If this paragraph doesn't make sense, don't worry, you can always get the right answer the original way.)