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Question Video: Understanding the Addition Rule in Probability Mathematics

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

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Video Transcript

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

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 𝐡 both occurring is π‘ž. This means the probability that 𝐴 occurs will be equal to 𝑝 plus π‘ž. Similarly, for event 𝐡, the probability of only 𝐡 occurring is π‘Ÿ, and the probability of 𝐡 and 𝐴 both occurring is π‘ž. That means the chances of event 𝐡 occurring will be π‘Ÿ plus π‘ž. Using this information and the Venn diagram, let’s try and find these expressions.

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

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

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

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

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