Question Video: Using the Addition Rule to Determine the Probability That at Least One of the Events Does Not Occur | Nagwa Question Video: Using the Addition Rule to Determine the Probability That at Least One of the Events Does Not Occur | Nagwa

# Question Video: Using the Addition Rule to Determine the Probability That at Least One of the Events Does Not Occur Mathematics • Second Year of Secondary School

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Suppose 𝐴 and 𝐵 are two events with probability 𝑃(𝐴) = 0.6 and 𝑃(𝐵) = 0.5. Given that 𝑃(𝐴 ∩ 𝐵) = 0.4, what is the probability that neither of the events occur?

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### Video Transcript

Suppose 𝐴 and 𝐵 are two events with probability 𝑃 of 𝐴 equals 0.6 and 𝑃 of 𝐵 equals 0.5. Given that the probability of 𝐴 intersection 𝐵 is 0.4, what is the probability that neither of the events occur?

In this question, we need to calculate the probability that neither of the events occur. As shown on the Venn diagram, this is the same as the complement of the union of events 𝐴 and 𝐵. As an event and its complement sum to one, we can therefore calculate the probability that neither event 𝐴 nor event 𝐵 occur by subtracting the probability of 𝐴 union 𝐵 from one. In this question, we are not given the probability of the union. However, we are given the probability of event 𝐴, the probability of event 𝐵, and the probability of the intersection of events 𝐴 and 𝐵. This means that we can begin by using the additive rule of probability, which states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.

Substituting in the values given, the right-hand side becomes 0.6 plus 0.5 minus 0.4. The probability of 𝐴 union 𝐵 is equal to 0.7. This is the area shaded in pink on our Venn diagram. We can now find a complement of this by subtracting 0.7 from one, which is equal to 0.3. We can therefore conclude that if the probability of 𝐴 is 0.6, the probability of 𝐵 is 0.5, and the probability of 𝐴 intersection 𝐵 is 0.4, then the probability that neither event 𝐴 nor event 𝐵 occurs is 0.3. This is the section outside of the circles on our Venn diagram.

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