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The definition of mutually exclusive events can be given in different ways. To begin with, two events are said to be mutually exclusive or disjoint if the occurrence of either one excludes the possibility of the other occurring . This means that they are events that cannot occur simultaneously . For example, when rolling a die only once, the result of landing on any of the six faces excludes it from landing on any of the other five. Thus, the event that lands 4 and the event that lands, say, 3, are mutually exclusive, since the die cannot land on both 4 and 3 at the same time.
On the other hand, in the field of probability it is said that two events are mutually exclusive as long as they do not share results with each other . This comes from the fact that, in probability, an event is considered as a set of possible results of an experiment. Different events can be defined that share or do not share results, and those that do not share results are considered mutually exclusive.
In more formal mathematical terms, and using set theory notation, events A and B will be mutually exclusive if their intersection is the empty set , that is, they do not intersect. In other words, A and B will be mutually exclusive as long as A ∩ B = Ø.
When are two events mutually exclusive?
In cases where logic does not tell us in advance whether two events are mutually exclusive, set theory and probability provide the solution. Here are three easy ways to determine, beyond a doubt, when two events are mutually exclusive or disjoint.
Observing the elements in each set
When two events contain a finite and small set of elements, it is very easy to determine whether or not they are disjoint, simply by checking whether or not they contain elements in common.
Example
Consider, for example, the experiment of rolling two dice simultaneously. Now let’s define the following two events:
- Let A be the event that the sum of the two dice is greater than or equal to 10.
- Let B be the event in which the sum of the two dice is exactly equal to 8.
It is easy to determine which results are included in each event. In the first, only the results (5,5); (5,6) and (6,6) result in a sum greater than or equal to 10. On the other hand, only the results (4,4); (5,3) and (6,2) yield 8. So now we can write, using set-theoretic symbology:
Since there are no common elements, the intersection is the empty set, and therefore the events are mutually exclusive.
Using Venn diagrams
Another very easy way to determine if two events are mutually exclusive is to represent them in a Venn diagram. In these diagrams, the sample space is represented by a rectangle (or other shape), while all events are represented as internal areas of the sample space.
In a Venn diagram, mutually exclusive events are easily recognized as those areas within the rectangle that do not touch or overlap.
By the probability of union
In some cases, the above two methods cannot be applied. An alternative way of checking whether or not two events are mutually exclusive is through probability. If the individual probabilities of each event are known, that is, P(A) and P(B), as well as the probability that one or the other event occurs, that is, P(AUB), then we know that two events are disjoint. if it is fulfilled that:
An alternative way is through the probability of intersection. Two events will be mutually exclusive as long as P(A ∩ B) = 0 .
Examples of mutually exclusive events
Simple events are always mutually exclusive
Simple events are those that contain a single result. When rolling a six-sided die, the event that it comes up 6 is a simple event, because it is made up only of the result 6. On the other hand, the event that it comes up even is not simple, since it is made up of three results, which They are 2, 4 and 6.
All simple events in an experiment will always be mutually exclusive.
Example
Suppose that a study determines the number of males born per week in a hospital. The sample space, S , for this experiment is
Some simple events would be:
As can be seen, since they do not have more than one result and they are all different, none of these events can share elements with another and, therefore, they will always be mutually exclusive.
Roll three dice simultaneously
Throwing three dice at the same time is an experiment that can have 36 different results, since the order of the dice does not matter: the results (1,2,3); (1,3,2); (2,1,3); (2,3,1); (3,1,2) and (3,2,1) all represent the same result.
Imagine that the following three events occur:
- A = event in which all dice give the same result.
- B = event in which only two dice give the same result.
- C = event in which all the dice give different results.
By common sense alone, it can be concluded that A, B, and C are all mutually exclusive events, since if all dice give the same result (event A occurs) it is impossible for only two to be the same and one different, or that all be different.
Card game
Imagine an experiment in which two cards are drawn at random from a deck of 52 poker cards. Now let’s define the following events:
- A = only red dots are drawn.
- B = only black dots are drawn.
These events are mutually exclusive, since if the cards are both red, they cannot both be black and vice versa.
Examples of events that are not mutually exclusive
Roll three dice simultaneously
Let’s take the same three dice experiment described above, but now define the following events:
- A = event where all dice are equal = {(1,1,1); (2,2,2); (3,3,3);…}
- B = event in which all dice are even = { (2,2,2); (2,2,4); (2,2,6)…}
By comparing the elements inside A and B, it is easy to see that there will be matches, and that the intersection of A and B will be:
Since the intersection is not the empty set, then these events are not disjoint.
Card game
Repeating the same experiment of drawing two cards from a deck, let us consider the following new events:
- A = at least one card is hearts.
- B = at least one card is a king.
In this case, whenever a King of Hearts is drawn, A and B are occurring at the same time. In fact, this is not the only outcome that happens, since if a King of Spades and an Ace of Hearts are drawn, A and B will also be occurring simultaneously. Therefore, A and B are not mutually exclusive events.
Importance and application of mutually exclusive events
In mathematics, the calculation of the probability of multiple events depends to a great degree on whether or not they are mutually exclusive. For example, one of the axioms of probability states that the union probability of several events is equal to the sum of the individual probability of each event if, and only if, all the events are mutually exclusive . In other words,
Only if A and B are disjoint or mutually exclusive events.
If they are not mutually exclusive, then the sum of the probabilities counts twice the probability of outcomes common to both events, ie the probability of intersection. For this reason, in these cases, the union probability is calculated in a different way:
For three events, A, B and C that are not mutually exclusive and that also intersect each other, things get even more complicated:
In this case, the probability of intersection of the three events, P( A ∩ B ∩ C) , must be added last, since it was subtracted three times by subtracting the intersections of the different pairs of events.
