Video Transcript
Denote by 𝐴 and 𝐵 two events in a sample space. Given that the probability of 𝐴 is equal to 0.45, probability of 𝐵 equals 0.6, and the probability of 𝐴 union 𝐵 is equal to 0.78, are the events 𝐴 and 𝐵 independent?
Well, the first thing we need to do is look at what independent events are. Independent events are events where the outcome of one event does not affect the outcome of the other. We can also say the probability of 𝐴 intersection 𝐵, so 𝐴 and 𝐵, is equal to the probability of 𝐴 multiplied by the probability of 𝐵. And this is only true if they are in fact independent.
Well, if you look at the information we’ve got, we got the probability of 𝐴. We’ve got the probability of 𝐵. However, we don’t have the probability of 𝐴 intersection 𝐵. We have 𝐴 union 𝐵. So what are we going to do?
Well, what we can use is one of our probability rules to help us work out what 𝐴 intersection 𝐵 is gonna be. And that 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. Well, you can think about why this would be the case using a Venn diagram cause we’ve got 𝐴 and 𝐵.
So if we think of the probability of 𝐴 union 𝐵 as the probability of 𝐴 or 𝐵, well then we can think of this as everything that we’ve shaded in here. However, if we wanted to make this, then this would be the same as plus the probability of 𝐵. But then what we’d have is the overlap in the middle. So therefore, we’d have to subtract this, which, we can see, would be the probability of 𝐴 intersection 𝐵 because this is in fact the intersection.
Okay, great, so now we’ve seen what we’ve got. Let’s get on and find out what our 𝐴 intersection 𝐵 is going to be. So substituting in our values, we’re gonna have 0.78 equals 0.45 plus 0.6 minus the probability of 𝐴 intersection 𝐵, which is gonna give 0.78 equals 1.05 minus the probability of 𝐴 intersection 𝐵. So therefore, the probability of 𝐴 intersection 𝐵 is gonna be equal to 1.05 minus 0.78, which is gonna give us the probability of 𝐴 intersection 𝐵 is equal to 0.27.
Okay, great, so now what we need to do is we want to work out whether our events are in fact independent. Well, to do that, what we need to do is work out if we get the same value for the probability of 𝐴 intersection 𝐵 when we multiply the probability of 𝐴 and the probability of 𝐵. So we’re gonna get the probability of 𝐴 intersection 𝐵 is equal to 0.45 multiplied by 0.6, which is equal to 0.27. So therefore, we can say that this is satisfied. So we can conclude that, yes, the events 𝐴 and 𝐵 are independent.
