Question Video: Understanding the Addition Rule in Probability Mathematics

The given Venn diagram represents the probabilities of events 𝐴 and 𝐵. Find an expression for 𝑃(𝐴 ⋃ 𝐵). Find an expression for 𝑃(𝐴) + 𝑃(𝐵). Find an expression for 𝑃(𝐴 ⋂ 𝐵). Hence, determine a formula for 𝑃(𝐴 ⋃ 𝐵) in terms of 𝑃(𝐴), 𝑃(𝐵), and 𝑃(𝐴 ⋂ 𝐵).

03:49

Video Transcript

The given Venn diagram represents the probabilities of events 𝐴 and 𝐵. Find an expression for the probability of 𝐴 union 𝐵. Find an expression for the probability of 𝐴 plus the probability of 𝐵. Find an expression for the probability of 𝐴 intersect 𝐵. Hence, determine a formula for the probability of 𝐴 union 𝐵 in terms of the probability of 𝐴, the probability of 𝐵, and the probability of 𝐴 intersect 𝐵.

First, let’s consider the information we’re given in the Venn diagram. We have probabilities for 𝐴, the probability of only 𝐴 occurring is 𝑝, and the probability of 𝐴 and 𝐵 both occurring is 𝑞. This means the probability that 𝐴 occurs will be equal to 𝑝 plus 𝑞. Similarly, for event 𝐵, the probability of only 𝐵 occurring is 𝑟, and the probability of 𝐵 and 𝐴 both occurring is 𝑞. That means the chances of event 𝐵 occurring will be 𝑟 plus 𝑞. Using this information and the Venn diagram, let’s try and find these expressions.

First, the probability of 𝐴 union 𝐵. This is the probability that 𝐴 or 𝐵 occurs. That would mean only 𝐴, only 𝐵, and the probability of both 𝐴 and 𝐵 occurring. Using the variables that represent those probabilities, that would mean 𝑝 plus 𝑞 plus 𝑟. If we want to write an expression for the probability of 𝐴 plus the probability of 𝐵, we’ve already said that the probability of 𝐴 is 𝑝 plus 𝑞 and the probability of 𝐵 is 𝑟 plus 𝑞. If we have 𝑞 plus 𝑞, we can simplify this to 𝑝 plus two 𝑞 plus 𝑟.

When we think about the probability of the intersection of 𝐴 and 𝐵, it is the probability that both 𝐴 and 𝐵 occur. And in a Venn diagram, that’s the overlapping space of 𝐴 and 𝐵. For this Venn diagram, the probability of both 𝐴 and 𝐵 occurring is 𝑞. Now, we want to write a formula for the probability of 𝐴 union 𝐵. And instead of using the 𝑝, 𝑞, and 𝑟, the terms need to be the probability of 𝐴, the probability of 𝐵, and the probability of the intersection of 𝐴 and 𝐵.

Remember, the probability of 𝐴 union 𝐵 is the probability that 𝐴 or 𝐵 occurs. Very often, we assume that that must be the probability of 𝐴 plus the probability of 𝐵. But here we see that that cannot possibly be true because the probability of 𝐴 or 𝐵 is 𝑝 plus 𝑞 plus 𝑟. And the probability of 𝐴 and 𝐵 is 𝑝 plus two 𝑞 plus 𝑟. We have an extra 𝑞-term. This is because if you add the probability of 𝐴 and the probability of 𝐵, you’re actually adding their intersection twice. To avoid this, we need to subtract the probability of the intersection of 𝐴 and 𝐵 from the probability of 𝐴 plus the probability of 𝐵. We’ll then say that we have 𝑝 plus 𝑞 plus 𝑟 is equal to 𝑝 plus 𝑞 plus 𝑟 plus 𝑞 minus 𝑞. We have removed that second addition of the overlap.

The formula for the probability of 𝐴 union 𝐵 in these terms is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of the intersection of 𝐴 and B.