# Question Video: Using the Addition Rule to Determine the Probability of Intersection of Two Events Mathematics

Denote by π΄ and π΅ two events with probabilities π(π΄) = 0.58 and π(π΅) = 0.2. Given that π(π΄ βͺ π΅) = 0.64, find π(π΄ β© π΅).

03:26

### Video Transcript

Denote by π΄ and π΅ two events with probabilities probability of π΄ is equal to 0.58 and the probability of π΅ is equal to 0.2. Given that the probability of π΄ union π΅ is equal to 0.64, find the probability of π΄ intersection π΅.

So the first thing I wanna do is actually show what the information weβve got means. So first of what we would have done is a small sketch of a Venn diagram. So weβve got π΄, the circle on the left-hand side and π΅ the circle on the right-hand side. Well, first of all, weβve got a bit of information that tells us the probability of π΄. So we can see that the probability of π΄ is equal to 0.58. And Iβve shown what the probability of π΄ means on our Venn diagram. And π΄ is the area here that Iβve coloured in. So Iβve shaded it in pink. Thatβs π΄ cause thatβs everything within the left-hand circle.

And then, next, what weβre told is that the probability of π΅ in occurring is equal to 0.2. What Iβve done is that Iβve shown this in another sketch. So you can see that that would be anything that occurs in this whole right-hand circle and that would be the probability of π΅. And then, weβre told that the probability of π΄ union π΅ is equal to 0.64. So what that means is that anything in π΄ or π΅ is gonna have a probability of 0.64. And we can see π΄ or π΅ is everything with the left-hand circle and everything with the right-hand circle. So that would be that area.

However, to solve the question, what weβre looking to find is a probability of π΄ intersection π΅. And what this means is π΄ and π΅. And this is the overlapping section in the middle. Iβve colored in blue. So how can we work this out? So we can see the probability of π΄ intersection π΅ would be equal to the probability of π΄ plus the probability of π΅. So we see from the diagrams that theyβll be the shaded areas of both circles, but then minus the probability of π΄ union π΅. Because if we took that away, that would give us the overlapping section, which would be our π΄ intersection π΅. So if we do that, we get 0.58 plus 0.2 minus 0.64, which will give us 0.78 minus 0.64 which would give us a final answer of the probability of π΄ intersection π΅ is equal to 0.14.

So great. I showed that using diagrams. And we can see exactly where it came from. And weβve got to our final answer. But there wouldβve been a quicker way we could have done it. And thatβs by using a formula. And what the formula tells us is that the probability of π΄ union π΅ is equal to the probability of π΄ plus the probability of π΅ minus the probability of π΄ intersection π΅, which we could have rearranged to the probability of π΄ intersection π΅ equals probability of π΄ plus the probability of π΅ minus the probability of π΄ union π΅. And we can do that by adding on the probability of π΄ intersection π΅ to both sides of the equation and subtracting the probability of π΄ union π΅. So we can get to our final answer either way.

And the final answer is the probability of π΄ and π΅ or the probability of π΄ intersection π΅ is equal to 0.14.

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