# Mutually Exclusive

Two events/choices that cannot occur at the same time

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## What is Mutually Exclusive?

Mutually exclusive is used to describe when two or more respective outcomes cannot occur simultaneously. If one of the results is chosen, all the other possible outcomes cannot be true at the same time. The most basic and commonly used example is a coin toss. With every toss of a coin, the outcome can either be heads or tails, but never both, making a coin toss a mutually exclusive event.

### Summary

• Choices/events are determined to be mutually exclusive when they cannot occur at the same time.
• Mutual exclusivity is taken into consideration for many business decisions, including investing and budget creation.
• Opportunity cost must be taken into consideration when making a decision regarding two or more mutually exclusive events.

### Mutually Exclusive and Probability

Mutual exclusivity is most commonly used in statistics and business decision-making. An event is deemed mutually exclusive if the occurrence of one outcome results in the non-occurrence of the other(s).

If an event is mutually exclusive, the probability of two of the possible results occurring is 0.

##### P(A and B) = 0

The probability of A and B occurring in a mutually exclusive event is 0. For example, the probability of pulling one card from a deck and it being a Jack and a Queen is zero (impossible). It cannot be both at the same time.

##### P(A and B) = P(A) + P(B)

For mutually exclusive events, the probability of A or B occurring is equal to the probability of A occurring plus the probability of B occurring. For example, the likelihood of pulling either a Jack or a Queen is equivalent to:

• Probability of a King is 1/13
• Probability of a Queen is 1/13

### Not Mutually Exclusive and Probability

Continuing with the theme of cards, we can look at an example that would not be mutually exclusive. The possibility of a Jack and a spade being pulled at the same time are not mutually exclusive because they can occur at the same time (Jack of spades).

When considering a Jack or spade, we use the formula of the probability of A plus the probability of B minus the probability and A and B.

Often, business decisions are mutually exclusive – although they should not be considered independent, as independent events have no significance on the viability of other options. Mutual exclusivity is inherently connected to opportunity cost, which is the cost of what a company or decision-maker must give up when they pursue one option over another. Mutually exclusive choices require the forfeit of one or more other options; hence, it is why opportunity cost is so tightly linked to the topic here.

Such dilemmas can be seen during capital budgeting and investment decision-making. All companies are limited by some resource – land, employees, capital, etc. – however, most often, the limiting factor is usually capital. Given such a limited resource, the company must select the most profitable project to pursue, ultimately resulting in the highest return on investment (ROI).

For example, let us assume a company’s investment budget is \$100,000, and three project options – A, B, and C. Projects A and B, are mutually exclusive, given the limiting resource of money. Therefore, if the company pursues project A, it cannot pursue project B or vice versa.

The business can choose to take on project A, or they can pursue projects B and C together. Project A comes with a cost of \$90,000 and a return of \$200,000, giving the project a net profit of \$110,000. Projects B and C cost \$80,000 and \$20,000 respectively, while returning \$170,000 and \$30,000, resulting in a net profit of \$100,000.

The opportunity cost of choosing project A is \$0, as it is the most profitable option, and no other decision will result in a better return. Choosing projects B and C would result in an opportunity cost of \$10,000 (\$110,000 – \$100,000), found using the formula below:

##### Opportunity Cost = Profit of Most Lucrative Project – Profit of Chosen Project

Therefore, we know that the company will choose to pursue Project A with zero opportunity cost and the highest return. Capital budgeting is one of many times that businesses analyze mutually exclusive events to find the optimal outcome to pursue.

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