What Is A Mutually Exclusive Event? And How To Analyse Them

Updated 30 September 2022

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In probability and statistics, two events are mutually exclusive when they cannot happen at the same time. In such situations, you can choose only one event and prevent the other one from occurring. Understanding everything about mutually exclusive events can help you better assess deal making and budgeting. In this article, we answer ‘What is a mutually exclusive event?', understand what independent events are and determine the steps required to analyse the probability of mutually exclusive events with some examples.

What Is A Mutually Exclusive Event?

The answer to the question, "What is a mutually exclusive event?" is that these are two or more events that cannot occur simultaneously. For instance, when tossing a coin, the outcome can be either tails or heads. You cannot toss a coin to get both heads and tails simultaneously. When two events are mutually exclusive or disjointed, the probability of both the events appearing simultaneously is zero. If C and D are two events, then:

P (C ∩ D) = 0

P = probability

C = first event

D = second event

0 = determines that the two events cannot take place at the same time

Some other examples of mutually exclusive events are:

Rolling a die might show any number from one to six. You cannot roll number 6 and number 2 simultaneously.
Walking forward and backwards is mutually exclusive because you cannot walk forward or backwards simultaneously.
Turning left or turning right is mutually exclusive because you cannot turn left or right simultaneously.
Drawing a black or diamond from a card deck is mutually exclusive because all diamonds are red cards.

What Are The Probability Rules Of Mutually Exclusive Events?

The probability rules of mutually exclusive events are:

Multiplication rule

You use the multiplication rule to find the probability of events occurring simultaneously. The rule multiplication rule states that:

P (C ∩ D) = 0

Since events ‘C' and ‘D' cannot occur simultaneously, their joint probability of occurrence is zero.

Addition rule

Using the addition rule, it becomes easy to calculate the probability of at least one event occurring. The addition rule states that:

P (C U D) = P (C) + P (D)

Example Of Calculating The Probability Of Mutually Exclusive Events

Here is an example of calculating the probability of mutually exclusive events:

What is the probability of a die showing a number 4 or 5?

Let P (4) be the probability of getting a 4

P (5) is the probability of getting a 5

P (4) = 1/6

P (5) = 1/6

P (C U D) = P (C) + P (D)

P (4 or 5) = P (4) + P (5) = 1/6 + 1/6 = 1/3

The probability of showing a 4 or 5 is 1/3.

What Are Independent Events?

An independent event occurs when one event does not affect the outcome of the others. When two events are mutually exclusive, they are not independent. Independent events cannot be mutually exclusive. For example, if you roll a die once, the result is a two. Rolling the die for the second time might cause two or any other number less than or equal to six. Either way, the outcome of rolling a die once does not affect the other.

When making business decisions on mutually exclusive events, understanding the outcome of each event can help decide which one benefits the business and its customers the most.

How To Analyse Mutually Exclusive Events

For deciding mutually exclusive events, determine the variables that affect each event. Follow these steps to analyse mutually exclusive events:

1. Calculate opportunity cost

Opportunity cost is a benefit that a business or investor misses when choosing one option over the other. Opportunity cost is the economic cost of the business. Calculating the opportunity cost is essential because it allows companies to use their resources and funds. By looking at the opportunity cost of an option, a business can understand which option provides a maximum return on their investment. The formula for calculating the opportunity cost is:

Opportunity cost = return on best forgone option (FO) – return on chosen option (CO)

Opportunity cost = FO – CO

Example: Reyansh works in a hospital as a doctor, making ₹20,00,000 per year. He starts his clinic, but he knows he would require a significant investment. Ryan might not see profits in the first few months of starting his clinic. Finally, Reyansh opened his clinic and earned ₹6,00,000 the first year and ₹12,00,000 the second year. These two events are mutually exclusive because Reyansh cannot simultaneously work in the hospital and his clinic.

If Reyansh had stayed at the hospital, he would have earned ₹40,00,000 in two years. Instead, he earned just ₹18,00,000, a 45% decrease. The opportunity cost for the first two years is:

Opportunity cost = 40,00,000 – 18,00,000 = ₹22,00,000

2. Calculate the time value of money

Time value of money (TVM) is a financial concept that states the value of money is worth more than the same amount in the future. This is true because you can invest the money you have right now and earn a return. This creates a larger amount of money in the future. To calculate the time value of money, it is essential to calculate:

Future value of money or FV: The future value of money answers whether the project's profit is worth the time it takes and the risk involved.
Present value of money or PV: The present value of money identifies the current value of the future sum of money. It determines whether investing somewhere else can cause more earnings.

Example: Rahul earns 10% as saving interest at a bank. His friend owed him money and offered to pay back ₹1,000 today or ₹1,040 next year from today. If he puts the ₹1,000 in the savings bank, he could earn ₹1,100 by the end of the next year. So, it is advisable for Rahul to take immediate repayment from his friend because the time value of ₹1,040 in a year is less than ₹1,100 today.

3. Calculate the net present value

Net present value or NPV is the present value of all the project's future cash flows compared to the initial investments. As the calculation of NPV considers time and value of money, businesses use it to compare projects and understand which project or investment results in the desired yield. Often, organisations use the NPV value to determine whether implementing a project returns positive results. The formula for calculating the NPV is:

NPV = [(cash flow) / (1+r)^t] – initial investment

r = discount rate or interest rate

t = period or number of periods during which the cash flow occurs

Another formula for calculating the NPV is:

NPV = Today's cash flow – Today's value of invested cash

A positive NPV shows a higher expected earning of an investment or project exceeding the initial investment cost. This means such a project would be profitable.

Example: An investor decides to invest ₹2,00,000 in Zoran Pharmaceuticals. The investment is likely to generate a cash flow of ₹2,50,000 over the next year. The rate of return is 15% per year.

NPV = 2,50,000 / (1 + 0.15) ^1 – 2,00,000 = ₹17,391

The NPV shows that the investment is profitable, the investor can make the investment.

4. Use the mutually exclusive and collectively exhaustive principle

After calculating the NPV, TVM and operating cash flow, focus on using the mutually exclusive and collectively exhaustive (MECE) principle. MECE is a grouping principle that separates items into various subsets as collectively exhaustive (CE) and mutually exclusive (ME). This systematic problem-solving framework helps a company understand every aspect of the investment or project before making a business decision.

Collectively exhaustive: A set of events is collectively exhaustive if at least one event is bound to occur. For instance, when you roll a die, it has to show any value from one to six and the numbers from one to six are collectively exhaustive.
Mutually exclusive: Two or more events that cannot occur simultaneously are mutually exclusive to each other.

Example: A leading FMCG company plans to add new products to its product line and the company is thinking between potato chips and potato wedges. Using the MECE principle, the company creates a list of factors that might affect their decision. They divide these factors into broader categories, such as cost and revenue and analyse these details:

Brand
Price
Competition
Investment
Time to market
Cost of materials

They break every component into a specific box for analysis and assign a variable to one of these ME categories. This helps in making strategic business and financial decisions.