Question Video: Using a Venn Diagram to Determine the Union of Two Events Mathematics

Video Transcript

Suppose that the probability of 𝐴 is equal to 0.3, the probability of 𝐵 is equal to 0.2, and the probability of 𝐴 intersect 𝐵 equals 0.1. Using a Venn diagram, find the probability of 𝐴 union 𝐵.

First, we should remember that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 or 𝐵 occurring. To draw a Venn diagram, we’ll need a circle that represents the probability of 𝐴 occurring and a circle for the probability of 𝐵 that overlaps the probability of 𝐴. And around these two circles, we draw a rectangle. This rectangle represents our total sample space. The space inside the rectangle but outside the circles 𝐴 and 𝐵 is the probability that 𝐴 or 𝐵 do not occur. The overlapping space between these two circles is the probability that both 𝐴 and 𝐵 occur. This is the probability of the intersection of 𝐴 and 𝐵.

We’ve already been told that this probability is 0.1. We also know the probability of 𝐴 occurring, and that is 0.3. However, this 0.3 will also include the 0.1 chance that both 𝐴 and 𝐵 occur. And that means the probability that 𝐴 occurs but 𝐵 doesn’t is 0.2. The probability of 𝐴 is the probability that only 𝐴 happens plus the probability that 𝐴 and 𝐵 both happen. When we’re looking at the probability of 𝐵, we know the total probability of 𝐵 is 0.2. 0.1 of that is the probability that both 𝐴 and 𝐵 occur. Then we have a remaining 0.1 probability that is the chance that only 𝐵 occurs.

To find the probability of 𝐴 or 𝐵 occurring, the probability of the union of 𝐴 and 𝐵, we need to add the probability that only 𝐴 occurs plus the probability that only 𝐵 occurs plus the probability that both 𝐴 and 𝐵 occur. For us, that means the probability of 𝐴 union 𝐵 under these conditions will be 0.2 plus 0.1 plus 0.1, which is 0.4.