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

Suppose that π(π΄) = 0.3, π(π΅) = 0.2, and π(π΄ β© π΅) = 0.1. Using a Venn diagram, find π(π΄ βͺ π΅).

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### Video Transcript

Suppose that the probability of π΄ is equal to 0.3, the probability of π΅ is equal to 0.2, and the probability of π΄ intersect π΅ equals 0.1. Using a Venn diagram, find the probability of π΄ union π΅.

First, we should remember that the probability of π΄ union π΅ is equal to the probability of π΄ or π΅ occurring. To draw a Venn diagram, weβll need a circle that represents the probability of π΄ occurring and a circle for the probability of π΅ that overlaps the probability of π΄. 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 π΄ and π΅ is the probability that π΄ or π΅ do not occur. The overlapping space between these two circles is the probability that both π΄ and π΅ occur. This is the probability of the intersection of π΄ and π΅.

Weβve already been told that this probability is 0.1. We also know the probability of π΄ occurring, and that is 0.3. However, this 0.3 will also include the 0.1 chance that both π΄ and π΅ occur. And that means the probability that π΄ occurs but π΅ doesnβt is 0.2. The probability of π΄ is the probability that only π΄ happens plus the probability that π΄ and π΅ both happen. When weβre looking at the probability of π΅, we know the total probability of π΅ is 0.2. 0.1 of that is the probability that both π΄ and π΅ occur. Then we have a remaining 0.1 probability that is the chance that only π΅ occurs.

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