Expected value is a statistical concept that helps us to figure out the value of a variable in the “average” case. Expected value tells us what to expect, which gives it many applications in real life.
So, what are some uses of expected value? Some uses of expected value include: business decisions, call centers, emergency room visits, future value of investments, insurance, lawsuits & settlements, rate of return on an investment (ROI), traffic patterns, store customers, and website traffic (CAC vs LTV).
Of course, expected value is just one tool that can help us to make decisions in business and in everyday life. Remember that an expected value may not be a possible outcome, but rather it is the outcome in the “average case” (over many trials).
In this article, we’ll take a look at 10 uses of expected value, along with some calculations to show how to apply this concept.
Let’s get started.
10 Uses Of Expected Value
Expected value is used in various scenarios, including:
- Business Decisions
- Call Centers
- Emergency Room Visits
- Future Value Of Investments
- Insurance
- Lawsuits & Settlements
- Rate Of Return On An Investment (ROI)
- Traffic Patterns
- Store Customers
- Website Traffic (CAC vs LTV)
Let’s take a closer look at each one, along with examples. We’ll start with call centers.
Business Decisions
Sometimes, a large business must decide whether to acquire a smaller business or to put the money to work elsewhere (an alternative investment).
Expected value can help us to make an informed decision when we face these scenarios. To use expected value, you must do two things:
- Assign probabilities for each outcome (either by calculation or estimation).
- Assign values for each outcome (either by calculation or estimation).
Example: Using Expected Value To Make A Business Decision
Let’s say that a large company has two options to invest:
- The first option is to spend $10 million to acquire a smaller competitor.
- The second option is to invest the $10 million into CDs that pay 3% per year, guaranteed.
Based on the competitor’s historical performance and your company’s expertise, your CFO outlines the following outcomes and probabilities:
- Scenario 1: The value of the competitor grows to $14 million in one year.
- Scenario 2: The value of the competitor remains exactly the same (at $10 million).
- Scenario 3: The value of the competitor falls to $8 million in one year.
The CFO thinks that there is a 20% chance of scenario 1, a 50% chance of scenario 2, and a 30% chance of scenario 3.
Then the expected value of the competitor in 1 year is:
- ($14 million)(0.2) + ($10 million)(0.5) + ($8 million)(0.3)
- =$2.8 million + $5 million + $2.4 million
- =$10.2 million
If the company spends $10 million to buy its competitor, the expected value in one year is $10.2 million (a gain of $200,000).
However, if the company invests the $10 million in CDs at 3%, the gain will be ($10 million)(0.03) = $300,000.
It seems that the better choice is to invest the money in a CD. Of course, a change in the probabilities or values for the competitor could change the calculation (and thus the action the company takes).
Call Centers
A company’s customer service call center will get a certain number of calls every minute and hour. These numbers will vary throughout the day, so staffing levels should change to reflect this.
If the company hires too many customer service agents, their employees may become bored with a lack of work. The company will also pay more than it needs to, which leads to higher costs and lower profit margins (or higher prices for customers).
If the company hires too few customer service agents, their employees will be overworked. The customers will also have longer wait times on the phone, and the company’s reputation may suffer.
As you can see, it’s important for the company to schedule enough workers, but not too many. Let’s take a look at an example to see how this might work.
Example: Using Expected Value To Staff A Call Center
Let’s say that in a given hour (noon to 1pm Eastern Standard Time), your company expects 17 people to phone the call center. Based on past experience, you know that it takes an agent an average of 10 minutes to help each customer.
This means that you expect to need 17*10 = 170 minutes of customer service during the hour from noon to 1pm. Since a single agent can only work 60 minutes in an hour, you will need 170 / 60 or about 3 agents on call during that hour.
Of course, more than 3 people could phone in at around the same time. In that case, the 4th, 5th, etc. people would be put on hold until an agent is available.
If you can minimize the time your customers are on hold, they will be much happier!
Emergency Room Visits
This scenario is similar to the call center example, but in this case, health and lives are at stake. As such, it might be wise to include a little more leeway after you find the expected value for the number of doctors or nurses you need standing by.
Future Value Of Investments
When deciding on which investment to make, expected value can help you to make sense of the opportunities in front of you – even when the outcomes are not clear at the time.
Using Expected Value To Analyze Investment Values
Let’s say that you have the option to invest $1 million in two companies. The values in 1 year, along with the probability of each, are given below:
[table for company A]
[table for company B]
The expected value of Company A in 1 year is:
- ($2 million)(0.1) + ($1.5 million)(0.7) + ($0.8 million)(0.2)
- =$0.2 million + $1.05 million + $0.16 million
- =$1.41 million
The expected value of Company B in 1 year is:
- ($3 million)(0.1) + ($1 million)(0.8) + ($0.5 million)(0.1)
- =$0.3 million + $0.8 million + $0.05 million
- =$1.15 million
Although the potential for $3 million with Company B is tempting, the expected value calculations tell us that in the long run, we are better off choosing investments like Company A.
In this case, we expect an extra $0.26 million ($260,000) by investing in Company A over Company B.
Insurance
An insurance company can calculate their expected claims liability by the formula
- (Claims Liability) = (Number Of Claims)(Average Payout Per Claim)
If an insurer can get a good estimate on the number of claims and the average payout per claim, then the claims liability estimate will be more accurate.
Example: Using Expected Value To Estimate Claims Liability
Let’s say that an insurance company expects the following number of claims and payouts:
- Auto: 150 claims, with an average payout of $12,000 per claim
- Home: 200 claims, with an average payout of $48,000 per claim
Then the claims liability estimates are as follows:
- Auto: (150 claims)($12,000 per claim) = $1,800,000
- Home: (200 claims)($48,000 per claim) = $9,600,000
So the insurance company’s estimate for total claims liability for auto and home is $1.8 million + $9.6 million = $11.4 million.
Lawsuits & Settlements
Lawyers can use expected value to determine if a case is worth pursuing. At the other end of the table, a company can use expected value to decide if it is better to fight a case or offer a settlement.
Example: Using Expected Value To Decide (Settle Or Fight)
Let’s say that a business is facing a lawsuit from a customer who claims that the company’s product harmed him. The company’s counsel believes that the case has little standing, but he lays out the possible outcomes and probabilities:
- Huge Loss: in this case, the customer wins $100 million from the company (2% chance).
- Minor Loss: in this case, the customer wins $1 million from the company (10% chance).
- Victory: in this case, the customer wins $0 (88% chance).
The expected value (in this case, a loss) for the company is:
- ($100 million)(0.02) + ($1 million)(0.10) + ($0)(0.88)
- =$2 million + $1 million + $0
- =$3 million
Although the company will most likely win the case (88% chance), there is still an expected loss of $3 million if they fight the case. Instead, they should seek to settle the matter out of court for less than $3 million.
One possible offer is $2 million. This saves the company $1 million on average, and it is twice what the man will get if he wins a $1 million award in court. Since he only has a 1% chance of winning the large award ($100 million), he may take the settlement if he is risk-averse.
Rate Of Return On An Investment (ROI)
Expected value can also help us to figure out the expected rate of return on an investment, based on how we think the company will do.
Example: Using Expected Value To Find ROI
After analyzing a company, you decide on the following rates of return and probabilities:
- +8% return, with a 20% probability
- +2% return, with a 70% probability
- -10% return, with a 10% probability
This gives us an expected rate of return as follows:
- (+8%)(0.2) + (+2%)(0.7) + (-10%)(0.1)
- =+1.6% + 1.4% – 1%
- =+2%
So, the expected value of the ROI (return on investment) is 2%. You could also compare this expected ROI to other companies to see which one you might want to invest in.
You could also split your money strategically by investing in multiple companies to give you a desired rate of return.
Traffic Patterns
The number of cars that pass over a bridge in an hour, day, month, and year will give us an indication of how much wear and tear to expect.
We can also use a Poisson distribution to find the probability for a given number of cars in a minute (or some other time period). This can help engineers to set up traffic lights (and timing) to ensure a smooth flow of traffic during peak times of the day.
Store Customers
This application of expected value is a combination of two examples: the traffic patterns and the call center. Instead of customer service agents, we might need to have cashiers to serve customers.
Example: Using Expected Value To Staff A Grocery Store
For a given hour of the day (3pm to 4pm), a grocery store expects 32 customers to enter the store. Based on historical data, they expect that the average customer will take 7 minutes to get through the checkout line.
This means that the store will need (32 customers)*(7 minutes per customer) = 214 minutes of checkout time from 3pm to 4pm. A clerk can only work 60 minutes in an hour, so you would need 4 clerks to work the cash registers, since 214 / 60 ~ 3.57 (round up to 4).
Website Traffic (CAC vs LTV)
Companies make several important decisions to remain profitable. One of these decisions is where and when to spend money on advertising.
A company can calculate out the return on investment for different advertising methods (Google Ads, Facebook Ads, YouTube ads, etc.) to help them make a decision. They can also compare the CAC (cost to acquire a customer) for a given advertising method with the LTV (lifetime value) of a customer.
Usually, it does not make sense to advertise if CAC is higher than LTV, unless the company is willing to lose money in the short term to increase market share in the long term.
Example: Using Expected Value To Decide Whether To Advertise
A company is currently advertising its diet products on a well-known food blogger’s website.
The company pays the blogger an advertising fee of $6,000 per month. The company studies its analytics and finds that the blogger’s website sends an average of 1,000 visitors per month to the company’s sales page.
Out of these visitors to the sales page, an average of 8% convert (that is, buy the company’s product). This means that an average of 8% x 1,000 = 80 paying customers per month come from the blogger’s website.
This means that the CAC (average cost to acquire a customer) through the food blogger’s website is:
- CAC = (total spent in acquiring customers) / (total customers acquired)
- CAC = $6,000 / 80
- CAC = $75
So, the company’s cost to acquire a customer is $75. If the company is making a profit of $100 per product sold, then they should continue advertising on the blogger’s website.
If the company is making a profit of $10 per product sold, then they should stop advertising on the blogger’s website (unless they want to temporarily lose money on every sale to increase market share).
At the “breakeven” point (profit of $75 per product sold), the company may wish to continue advertising to increase market share, even though it won’t bring any more immediate profit. The reason is that in the long term, the company may benefit from word of mouth advertising via a larger customer base.
Conclusion
Now you know a little more about expected value and how you might be able to use it in business or investment decisions.
You can read about the expected value of dice rolls in my article here.
You can learn about how probability is used in real life in this article.
You might also want to check out my article on the difference between probability & statistics.
I hope you found this article helpful. If so, please share it with someone who can use the information.
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~Jonathon