Video Transcript
If 𝐴 and 𝐵 are two independent events such that the probability of 𝐴 equals 0.2 and the probability of 𝐵 equals 0.6, determine the probability of 𝐴 union 𝐵.
Now, I read this notation in the question as 𝐴 union 𝐵, which in probability language means the probability of 𝐴 or 𝐵 occurring. In set notation, the set 𝐴 union 𝐵 is the set that contains all of the elements that are either in set 𝐴 or in set 𝐵 or in both. That’s the set that I’ve shaded in orange on the Venn diagram. There are standard formulae that we can apply to find the probability of 𝐴 union 𝐵, which we can also refer to as the probability of 𝐴 or 𝐵.
If the events 𝐴 and 𝐵 are mutually exclusive, which means there is no overlap between them, then we can find the probability of 𝐴 union 𝐵 by summing the individual probabilities of 𝐴 and 𝐵. If, however, the two events 𝐴 and 𝐵 are not mutually exclusive, which means that there is some overlap between them, then to find the probability of 𝐴 or 𝐵, we sum their individual probabilities. But, then, we need to subtract the probability of 𝐴 intersect 𝐵, which is the probability of 𝐴 and 𝐵 both happening.
If we look back at our Venn diagram, we can see why this is. And it’s because we’ve actually counted the section in the center of the Venn diagram, which is the intersection of 𝐴 and 𝐵, twice. We included it once in the circle for 𝐴, but then again in the circle for 𝐵. So as this section has been counted twice, we need to subtract this probability off again.
Now, we haven’t been told whether our events, 𝐴 and 𝐵, are mutually exclusive or not. So we’d need to use the second formula. However, we can always apply this formula anyway. As if the two events 𝐴 and 𝐵 are mutually exclusive, then there’s no overlap between them, which means the probability of their intersection would be equal to zero anyway. So we have then the probability of 𝐴 union 𝐵 is equal to 0.2, for the probability of 𝐴, plus 0.6, for the probability of 𝐵, minus the probability of 𝐴 intersect 𝐵. And so we need to calculate this probability.
Now, the probability of 𝐴 intersect 𝐵 means the probability of 𝐴 and 𝐵 both occurring. And we know that if two events are independent, then to find the probability of them both occurring, we’ll multiply their individual probabilities together. We’re told in the question that our events 𝐴 and 𝐵 are independent, which means we can apply this formula. So we know then that the probability of 𝐴 intersect 𝐵 is 0.2 multiplied by 0.6. Now this is equal to 0.12. And to see this, we can remember that two multiplied by six is equal to 12. And 0.2 is 10 times less than two. And 0.6 is 10 times less than six.
This means that the answer to the decimal calculation needs to be 100 times less than 12, which is 0.12. So we have that the probability of 𝐴 union 𝐵 is equal to 0.8 minus 0.12, which is equal to 0.68.
