If two events are independent, what is the probability that event $A$ OR event $B$ occurs?

$$A \ or \ B$$

in the case of independent events

Let's Look at the Venn Diagram

We don't care where our dart lands here. We'd be happy if it landed in only the blue part, only the yellow part, or right in the middle. All three of these cases make us happy.

Case 1: event $A$ occurs and $B$ doesn't
Case 2: event $B$ occurs and $A$ doesn't
Case 3: they both occur

The Formula

If we add up the areas of the two circles above, we would find the total area of all the places that make us happy. The formula is below. I want you to note that we're subtracting out an $AB$ term for some reason. Why are we doing that?

$$A \ or \ B = A + B - AB$$

Why are we subtracting the $AB$ term?

We subtract that $AB$ term, the probability of them BOTH happening (recall the previous mountain entry), because that area is actually being counted twice. That "overlap" area is part of the blue circle and the yellow circle, so when we combine them, we're actually counting that area twice. So we're not removing that area entirely when we subtract out the $AB$ term -- we're simply removing the double count.

If a problem asks us to specifically exclude that area, like in the problem below, look at how the formula changes:

If $A$ and $B$ are independent events, and the probability of event $A$ is $0.3$ and the probability of event $B$ is $0.5$, what is the probability that $A$ happens or $B$ happens, but not both?

$$A + B - 2AB$$

The probability of either event $A$ or $B$ occurring, but not both.

Note we subtract out the overlap area twice in order to completely remove it.