Two events are mutually exclusive

Mutually exclusive is a statistical term describing two or more events that cannot happen simultaneously. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other. For example, war and peace cannot coexist at the same time. This makes them mutually exclusive.

• Events are considered to be mutually exclusive when they cannot happen at the same time.
• The concept often comes up in the business world in the assessment of budgeting and dealmaking.
• If considering mutually exclusive options, a company must weigh the opportunity cost, or what it would be giving up by choosing each option.
• The time value of money (TVM) is often considered when deciding between two mutually exclusive choices.
• Not mutually exclusive means that two instances or outcomes can occur simultaneously, and one outcome does not limit the other from being possible.

Mutually exclusive events are events that can't both happen, but should not be considered independent events. Independent events have no impact on the viability of other options. For a basic example, consider the rolling of dice. You cannot roll both a five and a three simultaneously on a single die. However, you absolutely can roll a five and a three on two dice. Rolling a five and three simultaneously means this outcome is mutually exclusive. Rolling a five on one and a three on the other means they are not mutually exclusive outcomes.

When faced with a choice between mutually exclusive options, a company must consider the opportunity cost, which is what the company would be giving up to pursue each option. The concepts of opportunity cost and mutual exclusivity are inherently linked because each mutually exclusive option requires the sacrifice of whatever profits could have been generated by choosing the alternate option.

The time value of money (TVM) and other factors make mutually exclusive analysis a bit more complicated. For a more comprehensive comparison, companies use the net present value (NPV) and internal rate of return (IRR) formulas to mathematically determine which project is most beneficial when choosing between two or more mutually exclusive options.

The concept of mutual exclusivity is often applied in capital budgeting. Companies may have to choose between multiple projects that will add value to the company upon completion. Some of these projects are mutually exclusive.

For example, assume a company has a budget of $50,000 for expansion projects. If available Projects A and B each cost $40,000 and Project C costs only $10,000, then Projects A and B are mutually exclusive. If the company pursues A, it cannot also afford to pursue B and vice versa. Project C may be considered independent. Regardless of which other project is pursued, the company can still afford to pursue C as well. The acceptance of either A or B does not impact the viability of C, and the acceptance of C does not impact the viability of either of the other projects.

Moreover, when looking at opportunity costs, consider the analysis of Projects A and B. Assume that Project A has a potential return of $100,000, while Project B will only return $80,000. Since A and B are mutually exclusive, the opportunity cost of choosing B is equal to the profit of the most lucrative option (in this case, A) minus the profits generated by the selected option (B); that is, $100,000 - $80,000 = $20,000. Because option A is the most lucrative option, the opportunity cost of going for option A is $0.

In business, managers and directors often need to plan resource allocation. If a company is building a bridge and a skyscraper, and both projects require an extremely specialized piece of equipment and only one exists in the world, it would mean that these projects are mutually exclusive, as that piece of equipment cannot be used by both projects at the same time. This idea can be extended to consider specialized professionals, software systems (which cannot run both Mac and Windows), and allocated budgets.

To illustrate the difference between what is independent and what is mutually exclusive, consider the war and peace example from earlier. There could be war in France and peace in Italy. These are two independent nations and therefore each one could be in its own state of peace. However, there cannot be war in France and peace in France. Since they cannot coexist, that makes them mutually exclusive.

Typically, this involves budgeting and payments. If a company has $180 million to spend, it cannot spend that $180 million both by reinvesting in the business and offering bonuses to upper management. In this case, those two options are mutually exclusive. If the company can only retain licensing in a single country, that means they should not attempt to be licensed in two separate countries as they are mutually exclusive.

Things that are mutually exclusive are not able to occur simultaneously. In business, this is typically concerning the undertaking of projects or allocating a budget. If two things are not mutually exclusive, it means the existence and occurrence of one does not necessarily mean the other cannot coexist.

When you toss a coin, you either get heads or tails, but there is no other way you could get both results. This is an example of mutually exclusive events. In probability theory, two events are said to be mutually exclusive events if they cannot occur at the same time or simultaneously. In other words, mutually exclusive events are called disjoint events.

Further, if two events are considered disjoint events, then the probability of both events occurring at the same time will be zero. Let us learn more about this concept in this short lesson along with solved examples.

What Are Mutually Exclusive Events?

What is the meaning of mutually exclusive events? Mutually exclusive events are the events that cannot occur or happen at the same time. In other words, the probability of the events happening at the same time is zero. 

Example of Mutually Exclusive Events

A student wants to go to school. There are two paths; one that takes him to school and the other one that takes him home. Which path will he choose? He will choose one of the two paths. Obviously, he can't choose both at the same time. This is an example of a mutually exclusive events. 

How Do You Calculate Mutually Exclusive Events?

Mutually exclusive events are events that cannot occur or happen at the same time. The occurrence of mutually exclusive events at the same time is 0. If A and B are two mutually exclusive events in math, the probability of them both happening together is: P(A and B) = 0. The formula for calculating the probability of two mutually exclusive events is given below:

P(A or B) =  P(A) + P(B)

Do you know special symbols are used to show the relation between two sets: The two important relationships between two sets are the intersection of sets and union of sets.

Intersection of sets: The symbol used for the intersection is "\(\cap\)" and "and" is also used. If two sets are there say for example; A = {1, 2, 3} and B = {2, 3, 4}. Then A intersection B is represented as\( A\cap B\).

 A \(\cap\) B = {2, 3}

Union of sets: The symbol used for the union is "\(\cup\) " and "or" is also used. If two sets are there say for example; A = {1, 2, 3} and B = {2, 3, 4}. Then A union B is represented as \(A\cup B\).

A \(\cup\) B = {1, 2, 3, 4}

Probability of Disjoint (or) Mutually Exclusive Events

The probability of disjoint or mutually exclusive events A and B is written as the probability of the intersection of the events A and B. Probability of Disjoint (or) Mutually Exclusive Events = P ( A \(\cap \) B) = 0. In probability, the specific addition rule is valid when two events are mutually exclusive events. It states that the probability of either event occurring is the sum of probabilities of each event occurring. If A and B are said to be mutually exclusive events then the probability of an event A occurring or the probability of event B occurring is given as P(A) + P(B), ,

P (A U B) = P(A) + P(B)

Some of the examples of the mutually exclusive events are:

• When tossing a coin, the event of getting head and tail are mutually exclusive events. Because the probability of getting head and tail simultaneously is 0.
• In a six-sided die, the events “2” and “5” are mutually exclusive events. We cannot get both events 2 and 5 at the same time when we threw one die.
• In a deck of 52 cards, drawing a red card and drawing a club are mutually exclusive events because all the clubs are black.

If the events A and B are not mutually exclusive events, the probability of getting A or B is given as:

P (A U B) = P(A) + P(B) – P (A\(\cap \) B)

How Do You Show Mutually Exclusive Events?

We can use Venn diagrams to show mutually exclusive events. The figures shown below indicate mutually exclusive events and events that are not mutually exclusive events or non-mutually exclusive events. Note that there is no common element in mutually exclusive events.

Do Mutually Exclusive Events Add up to 1?

We know that mutually exclusive events cannot occur at the same time. The sum of the probability of mutually exclusive events can never be greater than 1 It is always less than 1, until and unless the same set of events are also exhaustive (at least one of them being true). In this case, the sum of their probability is exactly 1.

Mutually Exclusive Events Probability Rules

In probability theory, two events are mutually exclusive events or disjoint if they do not occur at the same time. A clear case is the set of results of a single coin toss, which can end in either heads or tails, but not for both. While tossing the coin, both outcomes are collectively exhaustive, which suggests that at least one of the consequences must happen, so these two events collectively exhaust all the possibilities.

Though, not all mutually exclusive events are commonly exhaustive. For example, the outcomes of 1 and 4 on rolling six-sided dice, are mutually exclusive events (both 1 and 4 cannot come as result at the same time) but are not collectively exhaustive (it can result in distinct outcomes such as 2,3,5,6). Further, from the definition of mutually exclusive events, the following rules for probability can be concluded.

• Addition Rule: P (A + B) = 1
• Subtraction Rule: P (A U B)’ = 0
• Multiplication Rule: P (A ∩ B) = 0

There are different varieties of events also. For instance, think of a coin that has a Head on both sides of the coin or a Tail on both sides. It doesn’t matter how many times you flip it, it will always occur Head (for the first coin) and Tail (for the second coin). If we check the sample space of such an experiment, it will be either { H } for the first coin and { T } for the second one. Such events have single point in the sample space and are called  “Simple Events”. Such kind of two sample events is always mutually exclusive events.

Conditional Probability for Mutually Exclusive Events

Conditional probability is stated as the probability of event A, given that another event B has occurred. For two independent events A and B, the conditional probability of event B given that  A has occurred is denoted by the expression P( B|A) and it is defined using the following equation.

P(B|A)= P (A ∩ B)/P(A)

Let us redefine the above equation using multiplication rule: P (A ∩ B) = 0

P(B|A)= 0/P(A)

So the conditional probability formula for mutually exclusive events is:

P (B | A) = 0

Important Notes 

Here are some important things to remember about mutually exclusive events:

1. The probability of an event that cannot happen is 0
2. The probability of an event that is certain to happen is 1
3.  The sum of the probabilities of all the elementary events of an experiment is 1
4. The probability of an event is greater than or equal to 0 and less than or equal to 1

1. Example 1: Daniel is trying to understand mutually exclusive events using dice. Help Daniel understand what is the probability of a dice showing 4 or 5?

   Solution:

   There are a total of 6 faces on a die, hence, the total number of outcomes will be 6

   The probability of a die showing 4 is P(4) = 1/6

   The probability of a die showing 5 isP(5) = 1/6

   The probability of getting 4 or 5 is = P(4 or 5)

   = P(4 or 5)

   = P (4) + P(5)

   = (1/6) + (1/6)

   = 1 + 1/ 6

   = 2/6

   = 1/3

   Answer: 1/3 will be the answer.

2. Example 2: Benny's teacher is teaching them about mutually exclusive events and gave him a deck of 52 cards and asked him to select a red card or a 6. Find the probability of selecting a red card or a 6
   Solution:

   The probability of getting a Red card = 26/52

   The probability of getting a 6 = 4/52

   The probability of getting both a Red and a 6 = 2/52

   P(R or 6) = P(R) + P(6) - P(R and 6)

   = (26/52) + (4/52) - (2/52)

   = (30-2/52)

   =28/52

   =7/13

   Answer: 7/13 will be the answer.

3. Example 3: Caroline noticed her mother trying to take out the fish to clean the fish tank. She asked her mother, "How many are males and how many are females?" Her mother replied that the tank contained 5 male fish and 8 female fish. What is the probability that the fish her mother takes out first is a male fish?

   Solution:

   This question can be solved easily by using the formula.

   Probability of an event = Number of possible outcomes/ Total no of favorable outcomes

   No. of male fish = 5 No. of female fish = 8

   Total no of fishes 

5+8 = 13

The probability that the fish are taken out is a male fish: No of male fish/ Total no of fishes

The probability that the fish are taken out is a male fish 5/13

Answer: Probability is 5/13.

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FAQs on Mutually Exclusive Events

Mutually exclusive events are a statistical term describing two or more events that cannot happen simultaneously. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other.

What Is an Example of Mutually Exclusive Events?

Mutually exclusive events are things that can't happen at the same time. For example, you can't run backward and forwards at the same time. The events “running forward” and “running backward” are mutually exclusive events. Tossing a coin can also give you this type of event.

What Does It Mean To Say Two Things Are Not Mutually Exclusive Events?

The two activities are said to be mutually exclusive events if one cannot exist when the other is true. Not mutually exclusive events means that they can take place at the same time. And we can say that "The two are not mutually exclusive."

What Is the Formula for Mutually Exclusive Events?

The formula for mutually exclusive events (they can't occur together), is that the (U) union of the two events must be the sum of both, i.e. 0.20 + 0.35 = 0.55.

How Do You Know if A and B are Mutually Exclusive Events?

A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A \(\cap\) B) = 0.

What Does Mutually Exclusive Events Mean in Probability?

In statistics and probability theory, two events are mutually exclusive events if they cannot occur at the same time. The simplest example of mutually exclusive events is a coin toss. A tossed coin outcome can be either head or tails, but both outcomes cannot occur simultaneously.

Are Dependent Events Mutually Exclusive Events?

Two mutually exclusive events are neither necessarily independent nor dependent.