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

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

03:26

Video Transcript

Denote by 𝐴 and 𝐡 two events with probabilities probability of 𝐴 is equal to 0.58 and the probability of 𝐡 is equal to 0.2. Given that the probability of 𝐴 union 𝐡 is equal to 0.64, find the probability of 𝐴 intersection 𝐡.

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

And then, next, what we’re told is that the probability of 𝐡 in occurring is equal to 0.2. What I’ve done is that I’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 𝐡. And then, we’re told that the probability of 𝐴 union 𝐡 is equal to 0.64. So what that means is that anything in 𝐴 or 𝐡 is gonna have a probability of 0.64. And we can see 𝐴 or 𝐡 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 we’re looking to find is a probability of 𝐴 intersection 𝐡. And what this means is 𝐴 and 𝐡. And this is the overlapping section in the middle. I’ve colored in blue. So how can we work this out? So we can see the probability of 𝐴 intersection 𝐡 would be equal to the probability of 𝐴 plus the probability of 𝐡. So we see from the diagrams that they’ll be the shaded areas of both circles, but then minus the probability of 𝐴 union 𝐡. Because if we took that away, that would give us the overlapping section, which would be our 𝐴 intersection 𝐡. 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 𝐴 intersection 𝐡 is equal to 0.14.

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

And the final answer is the probability of 𝐴 and 𝐡 or the probability of 𝐴 intersection 𝐡 is equal to 0.14.

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