𝐴 and 𝐵 are two independent events of a sample space of a random experiment. If the probability of 𝐴 equals 0.5 and the probability of 𝐵 equals 0.6, find the probability of 𝐴 union 𝐵.
This notation here, which I read as the probability of 𝐴 union 𝐵, means the probability of event 𝐴 or event 𝐵 occurring. It also includes the probability of both events occurring. There is a standard formula that we can apply in order to work this out. The probability of 𝐴 union 𝐵, or the probability of 𝐴 or 𝐵, is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersect 𝐵. That’s the probability of 𝐴 and 𝐵 both occurring.
We can use a Venn diagram to see where this formula comes from. Here is our Venn diagram with two intersecting circles to represent events 𝐴 or 𝐵. The probability of 𝐴 union 𝐵, or 𝐴 or 𝐵 happening, would be all of the area inside the two circles for 𝐴 and 𝐵. That’s all of the area now shaded in green. If we were to just add the pink and orange circles together for the probability of 𝐴 and the probability of 𝐵, we would actually include that section in the intersection, the middle of the Venn diagram, twice. Because it’s included once in the pink circle and once in the orange circle.
This is why we have to subtract the probability of 𝐴 intersect 𝐵, the intersection of the two circles, because it’s being counted twice. Now let’s have a look at the information given in the question. We can see that we’ve been told the probability of 𝐴 is 0.5. And also, the probability of 𝐵 is 0.6. So we have the first two probabilities we need. But what about the probability of 𝐴 intersect 𝐵?
Well, another one of our probability rules tells us that if two events are independent, then if we want to find the probability of them both happening, that’s the probability of their intersection, we can multiply their individual probabilities together. Our events 𝐴 and 𝐵 are independent because we’re told this in the question. So to find the probability of 𝐴 intersect 𝐵, we can multiply 0.5 by 0.6. That gives 0.30 which we can just right as 0.3.
So now we have all the values we need to substitute into our formula for the probability of 𝐴 union 𝐵. The probability of 𝐴 is 0.5. The probability of 𝐵 is 0.6. And the probability we’ve just calculated for 𝐴 intersect 𝐵 is 0.3. 0.5 plus 0.6 is 1.1, and then subtracting 0.3 gives 0.8. So we found that for the independent events 𝐴 and 𝐵 with the given probabilities, the probability of 𝐴 union 𝐵, which remember is the probability of 𝐴 or 𝐵 or both occurring, is 0.8.