Definition of Expected Value

What is an expected value?

In layman’s terms, the expected value is a calculation that serves as the best prediction of a value. In financial gobble-d-gook, it is the probability-weighted average value of all possible outcomes.

Why do I care?

Understanding the expected value of a possible future event allows us to make mathematically sound decisions. We can decide if we want to make an investment. We can assign a reasonable price for our services. We can prioritize requirements. Expected value is a calculation that should be used when calculating ROI.

How is expected value calculated?

Have you ever participated in a 50-50 raffle? This is a common fund-raising technique in the US for school groups, organizations, etc. The group sells raffle tickets for $1 each to people. Each ticket sold has a uniquely identifiable number, and a duplicate copy that the organization keeps, to be put into the drawing. After all of the tickets have been sold, the organization randomly picks a single ticket. The person who has the matching number on their raffle ticket wins half of the money collected by the group. The group keeps the other half.

People with good math intuition will guess that the expected value of each raffle ticket is $0.50. And they would be right.

Imagine 10 tickets were sold for $1 each. The winner of the raffle will take home $5 (half of ten), and everyone else loses. Each ticket has a 10% chance of being the winning ticket (worth $5), and a 90% chance of being a losing ticket (worth nothing).

To calculate the expected value, you would multiple the probability of each possible outcome by the value of that outcome, and then add the results.

The expected value of a $1 raffle ticket is (0.1 * $5) + (0.9 * $0) = $0.50.

If 1000 tickets were sold, the match becomes (0.001 * $500) + (0.999 * $0) = $0.50.

This approach uses a discrete probability distribution to model the potential outcomes. Each outcome is listed with a probability of occurance and a value if it occurs. Then those values are summed up. This is the most common approach used to calculate expected value.

Note that there is no limit on the number of probable outcomes. Imagine another contest with the following rules:

• Each ticket costs $10
• One person will win $1000
• Two people will win $500
• Everyone else will win $5 (they only get half of their money back).

If 1000 people play the game, then the expected value would be calculated as follows:

(0.001 * $1000) + (0.002 * $500) + (0.997 * $5) = $6.985

Not a very good bet, because the expected value will be less than the cost to play.

If only 200 people play, the expected value would be

(0.005 * $1000) + (0.01 * $500) + (0.985 * $5) = $14.985

This would be an excellent investment of your $10

Using expected value

We can now apply this expected value concept in our ROI calculations, when making decisions about which requirements to include in a release, which projects to undertake, etc.