MUTUALLY EXCLUSIVE EVENTS: PROPERTIES AND EXAMPLES - MATHS - 2025

Two events are said to be mutually exclusive, when both cannot occur simultaneously in the result of an experimentation. They are also known as incompatible events.

For example, when rolling a die, the possible outcomes can be separated such as: Odd or even numbers. Where each of these events excludes the other (An odd and even number cannot come out in turn).

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Returning to the example of the dice, only one face will be up and we will obtain an integer data between one and six. This is a simple event as it only has one possibility of outcome. All simple events are mutually exclusive by not admitting another event as a possibility.

What are mutually exclusive events?

They arise as a result of operations carried out in set theory, where groups of elements constituted in sets and sub-sets are grouped or demarcated according to relational factors; Union (U), intersection (∩) and complement (') among others.

They can be treated from different branches (mathematics, statistics, probability and logic among others…) but their conceptual composition will always be the same.

What are the events?

They are possibilities and events resulting from experimentation, capable of offering results in each of their iterations. The events generate the data to be recorded as elements of sets and sub-sets, the trends in these data are reason for study for probability.

Examples of events are:

• The coin pointed heads.
• The match resulted in a draw.
• The chemical reacted in 1.73 seconds.
• The speed at the maximum point was 30 m / s.
• The die marked the number 4.

Two mutually exclusive events can also be considered as complementary events, if they span the sample space with their union. Thus covering all the possibilities of an experiment.

For example, the experiment based on tossing a coin has two possibilities, heads or tails, where these results cover the entire sample space. These events are incompatible with each other and at the same time are collectively exhaustive.

Every dual element or variable of Boolean type is part of mutually exclusive events, this characteristic being the key to define its nature. The absence of something governs its state, until it is present and is no longer absent. The dualities of good or bad, right and wrong operate under the same principle. Where each possibility is defined by excluding the other.

Properties of mutually exclusive events:

1. A ∩ B = B ∩ A = ∅
2. If A = B 'are complementary events and AUB = S (Sample space)
3. P (A ∩ B) = 0; The probability of simultaneous occurrence of these events is zero

Resources such as the Venn diagram greatly facilitate the classification of mutually exclusive events among others , since it allows to fully visualize the magnitude of each set or subset.

The sets that do not have common events or are simply separated, will be considered as incompatible and mutually exclusive.

Example of mutually exclusive events

Unlike tossing a coin in the following example, events are treated from a non-experimental approach, in order to be able to identify the patterns of propositional logic in everyday events.

• The first, made up of males between the ages of 5 and 10, has 8 participants.
• The second, females between 5 and 10 years old, with 8 participants.
• The third, males between the ages of 10 and 15, with 12 participants.
• The fourth, females between the ages of 10 and 15, with 12 participants.
• The fifth, males between 15 and 20 years old, has 10 participants.
• The sixth group, made up of females between 15 and 20 years old, with 10 participants.

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1. Chess, a single event for all participants, both sexes and all ages.
2. Child gymkhana, both sexes up to 10 years old. One award for each gender
3. Women's soccer, for ages 10 to 20. A prize
4. Men's soccer, for ages between 10 and 20 years. A prize

• Sample space: 60 participants
• Number of iterations: 1
• It does not exclude any module from the camp.
• The participant's chances are to win the prize or not to win it. This makes each possibility mutually exclusive for all participants.
• Regardless of the individual qualities of the participants, the probability of success of each one is P (e) = 1/60.
• The probability that the winner is male or female is equal; P (v) = P (h) = 30/60 = 0.5 These events being mutually exclusive and complementary.

• Sample space: 18 participants
• Number of iterations: 2
• The third, fourth, fifth and sixth modules are excluded from this event.
• The first and second groups are complementary within the award. Because the union of both groups is equal to the sample space.
• Regardless of the individual qualities of the participants, the probability of success of each one is P (e) = 1/8
• The probability of having a male or female winner is 1 because an event will be held for each gender.

• Sample space: 22 participants
• Number of iterations: 1
• The first, second, third and fifth modules are excluded from this event.
• Regardless of the individual qualities of the participants, the probability of success of each one is P (e) = 1/2
• The probability of having a male winner is zero.
• The probability of having a female winner is one.

• Sample space: 22 participants
• Number of iterations: 1
• The first, second, fourth and sixth modules are excluded from this event.
• Regardless of the individual qualities of the participants, the probability of success of each one is P (e) = 1/2
• The probability of having a female winner is zero.
• The probability of having a male winner is one.

References

1. THE ROLE OF STATISTICAL METHODS IN COMPUTER SCIENCE AND BIOINFORMATICS. Irina Arhipova. Latvia University of Agriculture, Latvia.
2. Statistics and the Evaluation of Evidence for Forensic Scientists. Second Edition. Colin GG Aitken. School of Mathematics. The University of Edinburgh, UK
3. BASIC PROBABILITY THEORY, Robert B. Ash. Department of Mathematics. University of Illinois
4. Elementary STATISTICS. Tenth Edition. Mario F. Triola. Boston St.
5. Mathematics and Engineering in Computer Science. Christopher J. Van Wyk. Institute for Computer Sciences and Technology. National Bureau of Standards. Washington, DC 20234
6. Mathematics for Computer Science. Eric Lehman. Google Inc.

   F Thomson Leighton Department of Mathematics and the Computer Science and AI Laboratory, Massachussetts Institute of Technology; Akamai Technologies