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

Denote by 𝐴 and 𝐵 two events with probabilities 𝑃(𝐴) = 0.2 and 𝑃(𝐵) = 0.47. Given that 𝑃(𝐴 ∩ 𝐵) = 0.18, find 𝑃(𝐴 ∪ 𝐵).

03:22

Video Transcript

Denote by 𝐴 and 𝐵 two events with probabilities. The probability of 𝐴 equals 0.2, and the probability of 𝐵 equals 0.47. Given that the probability of 𝐴 intersection 𝐵 is equal to 0.18, find the probability of 𝐴 union 𝐵.

Well, the first thing I want to do is have a look at the information we’ve got and see what it means. So what I’ve done here is a sketch. And I’ve sketched a Venn diagram. And we have a circle on the left called 𝐴 and a circle on the right called 𝐵.

So the first bit of information we have is that the probability of 𝐴 is equal to 0.2. And I’ve shown that here on the Venn diagram, because in pink, I’ve shaded in the whole of circle 𝐴, because that’s what this would mean. Then next, we’re told that the probability of 𝐵 is equal to 0.47. I’ve shown this again here on the Venn diagram, this time by shading in the whole of the circle 𝐵. Then finally, we’re told that the probability of 𝐴 intersection 𝐵 is equal to 0.18. That’s what this bit of notation means that looks like an “n.” It means intersection. And as you can see on the diagram, I’ve shaded this in.

What intersection means is almost like “and,” like the probability of 𝐴 “and” 𝐵. So it’s the area that’s in both the circle 𝐴 and the circle 𝐵, also known as the crossover. However, to solve the problem, what we’re looking for is the probability of 𝐴 union 𝐵. What this means is anything that’s 𝐴 or 𝐵. And you can see that I’ve done this by shading in both the circle 𝐴 and the circle 𝐵 and the area that crosses over between them.

Well, to solve this problem, we have something called the addition rule. What this tells us is 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. So great, we can use this now to find the value that’s missing. But let’s see why this works.

Well, if we take a look at our diagrams, we can see that if we added the top two diagrams, so added the probability of 𝐴 and the probability of 𝐵. Then what we’d have is all of circle 𝐴 and all of circle 𝐵. However, as you can see, if we put them both together, we’d have two of these sections here. And these sections are the intersections.

And in fact, if we look at the bottom diagram, there’s obviously only one intersection. So what we’d have to do is subtract an intersection from our answer. Hence, while we get our addition rule that says that the probability of 𝐴 union 𝐵 equals the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.

Okay, great. So now let’s move on and solve the problem. So if we substitute in our values, we’re gonna get probability of 𝐴 union 𝐵 is equal to 0.2 plus 0.47 minus 0.18, which is gonna be equal to 0.67 minus 0.18. So therefore, we’ll get our final answer. And that is that the probability of 𝐴 union 𝐵 is equal to 0.49.

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