Video: Determining the Probability of the Difference of Two Events

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃(𝐴) = 5/7 and 𝑃(𝐵) = 4/7. Given that 𝑃(𝐴 ∪ 𝐵) = 6/7, determine 𝑃(𝐴 − 𝐵).

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Video Transcript

Suppose 𝐴 and 𝐵 are two events with probabilities the probability of 𝐴 equals five-sevenths and the probability of 𝐵 equals four-sevenths. Given that the probability of 𝐴 union 𝐵 is equal to six-sevenths, determine the probability of 𝐴 minus 𝐵.

So the first thing I’m gonna do is show what our information means using some sketches. So I’m gonna start first of all with the probability of 𝐴. And we have the probability of 𝐴 is equal to five-sevenths. And the probability of 𝐴 is all of the left circle which I’ve shaded in with pink in our Venn diagram. So that would represent the area which is the probability of 𝐴.

And then we have the probability of 𝐵 which is equal to four-sevenths. And the probability of 𝐵 is represented here with the right-hand circle which I’ve shaded here in pink. And finally, we’ve got the probability of 𝐴 union 𝐵, which is equal to six-sevenths.

And this U shape in the notation means union. And what this means is everything that is in 𝐴 or 𝐵. So I’ve shown that here because what we’ve got is both the circles shaded. Right, so that’s all the information that we have. But what are we trying to look for?

Well, what we’re trying to find is the probability of 𝐴 minus 𝐵. Well, let’s take a look at what this would mean. So if we did the probability of 𝐴 minus the probability of 𝐵, we’d have this circle, which is 𝐴. But then what we’d want to do is subtract circle which I’ve colored in, which is 𝐵. Well, this would mean that we’d remove this middle section here. And what we’d be left with is the area that I’ve shaded across in blue, which is the probability of 𝐴 minus 𝐵. And we remove this intersection because that was the only bit of 𝐴 that was also 𝐵. Okay, great. But how do we calculate this?

Well, what we have is a rule. And that’s the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. And this is what we’ve shown. Because 𝐴 intersection 𝐵 is the part in the middle. So the bit that’s in the middle of both far circles, the overlap.

So now we’ve got this. Let’s work it out. But hold on, there is one problem. We don’t know the probability of 𝐴 intersection 𝐵. So what we’re gonna have to use is another rule. And this rule 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. And what we can do is we can rearrange this to find the probability of 𝐴 intersection 𝐵.

So when we substitute in the values we know, we’re gonna get six-sevenths equals five-sevenths plus four-sevenths minus the probability of 𝐴 intersection 𝐵. So therefore, we’re gonna get six-sevenths equals nine-sevenths minus the probability of 𝐴 intersection 𝐵. So therefore, we can see that the probability of 𝐴 intersection 𝐵 is gonna be equal to three-sevenths. I’ll just work that out because if we got six-sevenths is equal to nine-sevenths minus something.

Well, nine-sevenths minus three-sevenths gives us six-sevenths. This is how this works. Or we could’ve rearranged it as we would with any equation and added the probability of 𝐴 intersection 𝐵 onto both sides and then subtracted six-sevenths from both sides.

So now what we’re gonna do is use our top rule to find the probability of 𝐴 minus 𝐵. So then what we’re gonna get is the probability of 𝐴 minus 𝐵 is gonna be equal to five-sevenths minus three-sevenths. And that’s because it’s probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. So therefore, we can say that we’ve solved the problem. And the probability of 𝐴 minus 𝐵 is equal to two-sevenths.

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