### Video Transcript

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

First, letβs consider the information weβre given in the Venn diagram. We have probabilities for π΄, the probability of only π΄ occurring is π, and the probability of π΄ and π΅ both occurring is π. This means the probability that π΄ occurs will be equal to π plus π. Similarly, for event π΅, the probability of only π΅ occurring is π, and the probability of π΅ and π΄ both occurring is π. That means the chances of event π΅ occurring will be π plus π. Using this information and the Venn diagram, letβs try and find these expressions.

First, the probability of π΄ union π΅. This is the probability that π΄ or π΅ occurs. That would mean only π΄, only π΅, and the probability of both π΄ and π΅ occurring. Using the variables that represent those probabilities, that would mean π plus π plus π. If we want to write an expression for the probability of π΄ plus the probability of π΅, weβve already said that the probability of π΄ is π plus π and the probability of π΅ is π plus π. If we have π plus π, we can simplify this to π plus two π plus π.

When we think about the probability of the intersection of π΄ and π΅, it is the probability that both π΄ and π΅ occur. And in a Venn diagram, thatβs the overlapping space of π΄ and π΅. For this Venn diagram, the probability of both π΄ and π΅ occurring is π. Now, we want to write a formula for the probability of π΄ union π΅. And instead of using the π, π, and π, the terms need to be the probability of π΄, the probability of π΅, and the probability of the intersection of π΄ and π΅.

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

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