A certain action can be performed in 𝑚 different ways. A second action, which is mutually exclusive of the first, can be performed in 𝑛 different ways. Write an expression for the number of ways to perform either the first action or the second action.
Let’s begin by reminding ourselves what we mean when we talk about two events being mutually exclusive. In the most basic of terms, two events are said to be mutually exclusive if they cannot happen at the same time. For example, suppose we flip a coin. The coin can land on heads or tails. The event the coin lands on heads is mutually exclusive to the event the coin lands on tails. These things cannot happen at the same time.
And it can be helpful to think about what that might look like in a Venn diagram. We’re used to a Venn diagram showing two events 𝐴 and 𝐵 looking like this. But of course if 𝐴 and 𝐵 are mutually exclusive, they cannot happen at the same time. And this means this overlap will be the empty set. In other words, the overlap can contain no elements. And so we might represent a Venn diagram which contains mutually exclusive events 𝐴 and 𝐵 as shown.
Now, let’s link this to our question. We’re told that a certain action can be performed in 𝑚 different ways. Let’s define this certain action to be 𝐴. Then, the number of elements in set 𝐴 is equal to 𝑚. Similarly, another action which is mutually exclusive of the first can be performed in 𝑛 different ways. Let’s define this action or event to be 𝐵. Then, the number of elements in set 𝐵 is 𝑛. We want to find the number of ways to perform either the first action or the second action. In other words, how many ways are there to choose an outcome from 𝐴 or 𝐵?
Well, in fact, we just find the sum of 𝑚 and 𝑛, meaning that there are a total of 𝑚 plus 𝑛 ways to perform either the first action or the second. But of course we could have saved ourselves some time here. This is essentially the addition rule for two events. If 𝐴 and 𝐵 are mutually exclusive events, where there are 𝑚 distinct outcomes of event 𝐴 and 𝑛 distinct outcomes of event 𝐵, then there are 𝑚 plus 𝑛 distinct outcomes from either 𝐴 or 𝐵.