# Operations with events

Event operations are event union, event intersection, and event difference.

Event operations are a fundamental part of the introduction to probability theory. They offer a framework for operating with sets. In the same way that we can operate with other types of elements, we can also do it with probabilities.

Within the operations with events there are several that are worth knowing. All of them are developed in our dictionary. Developed, explained and with resolved examples.

## Types of event operations

To simplify the explanation, we will assume that we have two events A and B.

• Union of events:The union of events is characterized by solving the question: What is the probability that A or B will come out?
• Eventintersection : The event intersection, on the other hand, answers the question: What is the probability that A and B will both come out?
• Eventdifference : The event difference can be normal or symmetric. The normal difference answers the question: What is the probability that A will exit and B will not exit? Meanwhile, the symmetric difference answers the question: What is the probability that A or B will come out, but not both at the same time?

Each of these operations has some properties. It is important to know these properties to have a statistical basis that allows us to learn more advanced concepts.

## Examples of event operations

Since each concept is developed individually, in what follows we will simply give an example with its result. That is, to see the explanation it is recommended to access each concept:

We have three events: A, B and C. Each of them has a probability of happening that is manifested below:

P (A): 0.5 P (B): 0.6 P (C): 0.1

P (AUC) : 0.3 and P (A ∩ B): 0.2

We will denote the complementary of B by *

Considering that A and B are not disjoint, what is the probability of the union?

P (AUB) = P (A) + P (B) – P (A ∩ B)

P (AUB) = 0.5 + 0.6 – 0.2 = 0.9

The probability of the union of A and B is 0.9. Or as a percentage, the probability is 90%.

Now, let’s look at an example of event intersection. Taking into account that A and C are not disjoint events, what is the probability of the intersection of A and C?

P (A ∩ C) = P (A) + P (B) – P (AUC)

P (A ∩ C) = 0.5 + 0.6 – 0.3 = 0.8

The probability of the intersection between A and C occurring is 0.8. That is, the probability that A and C occur at the same time is 80%.

Finally, let’s look at an example of normal event difference. What is the probability that A will occur and B will not occur?

P (A – B) = P (A ∩ B * ) = P (A) – P (A ∩ B)

P (A – B) = 0.5 – 0.2 = 0.3

The probability of the difference of events A and B (in that order) is 0.3. That is, the probability that A occurs and B does not occur is 30%. ##### byAbdullah Sam
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