Question Video: Using the Addition Rule to Determine the Probability of an Event | Nagwa Question Video: Using the Addition Rule to Determine the Probability of an Event | Nagwa

# Question Video: Using the Addition Rule to Determine the Probability of an Event Mathematics • Third Year of Preparatory School

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Suppose that 𝐴 and 𝐵 are two events. Given that 𝑃(B) = 0.06, 𝑃(A ∪ 𝐵) = 0.09, and 𝐵 ⊂ 𝐴, determine 𝑃(A).

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### Video Transcript

Suppose that 𝐴 and 𝐵 are two events. Given that the probability of 𝐵 is 0.06, the probability of 𝐴 union 𝐵 is 0.09, and 𝐵 is a subset of 𝐴, determine the probability of 𝐴.

We will begin this question by drawing a Venn diagram to model the situation. We are told that 𝐵 is a subset of 𝐴, which means that every element in 𝐵 is also contained in 𝐴. The question tells us that the probability of 𝐵 occurring is 0.06. And it also says that the probability of 𝐴 union 𝐵 is 0.09. This is the probability that 𝐴 occurs or 𝐵 occurs or they both occur.

To determine the probability of 𝐴, we recall one of our probability rules, which should be clear from the Venn diagram. If 𝐵 is a subset of 𝐴, then the probability of 𝐴 is equal to the probability of 𝐴 union 𝐵. The probability of 𝐴 is therefore equal to 0.09.

Whilst it is not required in this question, we could complete the Venn diagram as follows. Firstly, the probability of only 𝐴 occurring, written 𝑃 of 𝐴 minus 𝐵, is equal to 0.09 minus 0.06, which is equal to 0.03. The probability of neither 𝐴 nor 𝐵 occurring is one minus the probability of their union, in this case one minus 0.09. This is equal to 0.91. We can check this by ensuring that all the values in the Venn diagram sum to one.

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