7 junio, 2024

Mutually non-exclusive events: what they are, properties and examples

What are mutually non-exclusive events?

are considered mutually non-exclusive events all those events that have the capacity to occur simultaneously in an experimentation. The occurrence of any of them does not imply the exclusion of the other.

Unlike their logical counterpart, the mutually exclusive events, the intersection between these elements is different from a vacuum. This is:

A ∩ B = B ∩ A ≠ ∅

Because the possibility of simultaneity between outcomes is handled, mutually non-exclusive events require more than one iteration to cover probabilistic studies.

What are mutually non-exclusive events?

In probability, two types of eventualities are handled: the occurrence and non-occurrence of the event. The binary quantitative values ​​are 0 and 1. Complementary events are part of relationships between events, based on their characteristics and particularities, which can differentiate or relate them to each other.

In this way, the probabilistic values ​​cover the interval [ 0 , 1 ] varying its occurrence parameters depending on the factor sought in the experimentation.

Two mutually non-exclusive events cannot be complementary, because there must be a set formed by the intersection of both, whose elements are different from the void. Which does not meet the definition of plugin.

What are the events?

They are possibilities and events resulting from experimentation, capable of offering results in each of its iterations. The events generate the data to be recorded as elements of sets and subsets, and the trends that appear in these data are the subject of study for probability.

Mutually non-exclusive properties of events

Let A and B be two mutually non-exclusive events in the sample space S.

A ∩ B ≠ ∅ and the probability of occurrence of their intersection is P [ A ∩ B ]

P [ A U B ] =P [ A ] +P [ B ] –P [ A ∩ B ]

This is the probability that one event or another will occur. Due to the existence of common elements, the intersection must be subtracted so as not to add twice.

There are tools in set theory that make working with mutually non-exclusive events much easier.

The Venn diagram, among them, defines the sample space as the universe set, defining each set and subset within it. It is intuitive to find the intersections, unions and complements that are required in the study.

Examples of Mutually Non-Exclusive Events

Let it rain and let there be traffic on the highways. Both events can occur at the same time, as rainy weather can affect traffic. A student can study and get a good grade on an exam. If a person studies well, they are more likely to get a high grade. If you eat a lot, you are likely to gain weight. When you exercise, your physical condition improves. Roll a dice and get an even number or a prime number. When rolling a dice, it is possible to get a 2 (even number) or a 3 (prime number), since 2 is an even and prime number. If a card is drawn from a deck, it may be a red and odd numbered card. Choose a food for dinner that is a healthy option or a low-calorie option. Some foods can be healthy and low in calories at the same time. When you have a cold, it is highly likely that you also have nasal congestion. Both symptoms can be present simultaneously during a common cold. Taking a test and feeling nervous. Test anxiety may be present while taking the test. Buy a lottery ticket and not win the jackpot. It is possible to buy a lottery ticket and still not win the jackpot, but still have a chance to win a smaller prize.


Colin GG Statistics and the Evaluation of Evidence for Forensic Scientists. Second Edition. School of Mathematics. The University of Edinburgh.
Mario F. Triola. Elementary STATISTICS. Tenth Edition. Boston St.

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