Question Video: Using the Addition Rule to Determine the Probability of Intersection of Two Events | Nagwa Question Video: Using the Addition Rule to Determine the Probability of Intersection of Two Events | Nagwa

# Question Video: Using the Addition Rule to Determine the Probability of Intersection of Two Events Mathematics • Third Year of Preparatory School

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Denote by 𝐴 and 𝐵 two events with probabilities 𝑃(𝐴) = 0.58 and 𝑃(𝐵) = 0.2. Given that 𝑃(𝐴 ∪ 𝐵) = 0.64, find 𝑃(𝐴 ∩ 𝐵).

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### Video Transcript

Denote by 𝐴 and 𝐵 two events with probabilities the probability of 𝐴 equals 0.58 and the probability of 𝐵 equals 0.2. Given that the probability of 𝐴 union 𝐵 equals 0.64, find the probability of 𝐴 intersects 𝐵.

In this question, we are given the probabilities of two events 𝐴 and 𝐵, as well as the probability of the union of these events, that is, the probability of either event occurring. We want to use these probabilities to find the probability of the intersection of these events, that is, the probability of both events occurring.

We can do this by noting that we are given three of the four probabilities in the addition rule for probability and asked to find the fourth probability. We can recall that the addition rule for probability tells us that for any events 𝐴 and 𝐵, the probability of either event occurring, 𝑃 of 𝐴 union 𝐵, is equal to the probability of 𝐴 occurring added to the probability of 𝐵 occurring minus the probability of both events occurring, that is the probability of 𝐴 intersects 𝐵.

We can substitute the given probabilities into the addition rule for probability to obtain 0.64 is equal to 0.58 plus 0.2 minus the probability of 𝐴 intersects 𝐵. We can then rearrange this equation to solve for the unknown probability. We have 𝑃 of 𝐴 intersects 𝐵 is equal to 0.58 plus 0.2 minus 0.64. We can then evaluate this expression to obtain that the probability of both events 𝐴 and 𝐵 occurring is 0.14.

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