LESSON: The Addition Rule

In this lesson we will be considering "compound events". Remember that a compound event is any event combining two or more simple events. 

We've worked with the Addition Rule for probability already when we learned what this symbol meant, $$\bigcup$$. It means "or". When calculating a compound probability with the $$\bigcup$$, we are essentially adding the probability of both simple probabilities together. 

If our compound events are disjoint, meaning they cannot occur together, then this calculation is simple.

If our compound events are not disjoint, then that means the events can occur together and we need to be careful how we add the probabilities together.

In the upcoming lesson video you will see an example of using the Addition Rule for a compound event that is not disjoint. We won't formalize the definition for the addition rule just yet, but it's important to understand the sprit of the rule and how to apply it to a situation intuitively.

The figures below provide a visual illustration of the Addition Rule. Remind yourself that $$\bigcup$$ means "OR" and $$\bigcap$$ means "AND".

For this venn diagram that represents overlapping events, we can see that the Probability of A or B equals the probability of of A plus the probability of B minus the probability of A and B. This venn diagram shows that the addition of the areas of the two circles alone will cause double counting of the overlapping area. This is the basic concept that underlies the addition rule.

When we have overlapping events, the Addition Rule states: $$P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B)$$

The next Venn diagram shows two events that are disjoint, meaning the two events do not overlap. In this situation there is no space where $$P(A \bigcap B)$$ exists. Therefore it is not necessary to subtract it because it's equal to 0!

When we have disjoint events, the addition rule states: $$P(A \bigcup B) = P(A) + P(B)$$

It's helpful to see an example of how to use the Addition Rule once you've got a basic understanding of the formula.

In this lesson video I'm going to use a two way table to find probabilities instead of a Venn diagram to organize our information.

Keep in mind the general process for solving probabilities that involve using the addition rule:

• Use this formula for disjoint events: $$P(A \bigcup B) = P(A) + P(B)$$
  
• Use this formula for events that are not disjoint: $$P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B)$$