Expected value formula for decisions, with a worked example
The essentials in 30 seconds
- The expected value formula is EV = p1 × x1 + p2 × x2 + … + pn × xn, with probabilities that sum to 1.
- In the marketing example, the campaign has an expected value of 225, the pilot 200, and the partnership 180 (US$ thousands).
- The campaign carries a standard deviation of about 249, and the partnership 0.
- Cutting the strong case to 15 percent drops the campaign to 185.
- The EQOS Asia Prediction pillar projects 3 trajectories (EQOS Asia method page, as of October 2026).
The expected value formula multiplies each possible payoff by its probability and adds the results. It condenses an uncertain option into one number that can sit next to other options. This page writes the formula, applies it to three options with illustrative figures, and adds the spread, the probabilities, and the time value, as of October 2026. It extends the guide on how to use expected value in a decision.
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What is the expected value formula for a decision?
The formula reads EV = p1 × x1 + p2 × x2 + … + pn × xn, where each x is a payoff and each p is the probability of that payoff. The probabilities of all outcomes of one option must sum to 1, or 100 percent. If they sum to less, an outcome is missing, and the result understates or overstates the option.
Payoffs can be positive or negative, so a loss simply enters with a minus sign. The result is a long-run average: it tells what the option returns if the same situation were repeated many times. For a one-off decision it works as a fair price for the option, a figure to compare with the price of the alternatives.
How do you apply the expected value formula step by step?
Four steps apply the formula. List the outcomes of each option so that they cover every case without overlap. Write the payoff of each outcome net of the cost of the action, in one currency and one period. Assign a probability to each outcome, with the total at 1. Multiply, add, and compare the results across options.
Keep the outcomes few. Three to five outcomes per option capture the shape of the risk, and more outcomes add effort without insight. Write a one-line story for each outcome, and name the outcomes in words, such as strong, normal, and weak, so that a reader checks the story behind each number before checking the arithmetic.
What does the expected value formula return for three options?
Take three ways to spend a marketing budget, in US$ thousands, all figures illustrative. A broad campaign has a 25 percent chance of a strong result worth 600, a 50 percent chance of a normal result worth 200, and a 25 percent chance of a loss of 100. Its expected value is 150 + 100 − 25, or 225.
A partnership pays a fixed fee, so its single outcome of 180 has probability 1 and an expected value of 180. A small pilot has a 50 percent chance of 360 and a 50 percent chance of 40, which gives 180 + 20, or 200. The campaign leads on average, ahead of the pilot by 25 and the partnership by 45.
What does the expected value formula leave out about spread?
Two options with the same average can differ widely in risk. The standard deviation measures the spread around the expected value. For the campaign, the deviations from 225 are 375, minus 25, and minus 325. Squaring, weighting by probability, and adding gives a variance of 61,875, and its square root is a standard deviation of about 249.
The partnership has a standard deviation of 0, since its payoff is fixed. The campaign leads by 45 on average and carries a loss of 100 in one case out of 4. Showing the worst case and the spread beside the expected value lets the decision maker choose between a higher average and a steadier path.
Where do the probabilities in an expected value calculation come from?
Probabilities come from three places: the history of similar cases, data from a pilot, and the reasoned judgement of people who know the market. Start from the base rate. If 1 in 4 comparable campaigns produced a strong result, 25 percent is a defensible starting point, and the team moves away from it only for a stated reason.
Then test the estimate. Change the probability of the strong case from 25 to 15 percent and move the 10 points to the normal case. The campaign falls to 0.15 × 600 + 0.60 × 200 − 0.25 × 100, which equals 90 + 120 − 25, or 185. The pilot, at 200, now leads, and the team knows the choice depends on that single estimate. A sensitivity test of this kind takes one extra row and shows which input deserves better data.
How does the timing of payoffs affect the expected value?
Payoffs that arrive in different periods cannot be added directly. Discount each payoff to today with a rate that reflects the cost of capital. At an illustrative rate of 8 percent, an expected value of 225 received in 12 months is worth 225 divided by 1.08, or about 208 today.
Apply the same rule to every option. The partnership pays its fee of 180 in the same year, so its present value is about 167. The ranking is unchanged here, and the gap shrinks to about 42. When one option pays in year 1 and another in year 3, the discount can reverse the order.
How does the EQOS Asia decision engine extend the expected value formula?
The formula prices a few outcomes with probabilities that people supply. The EQOS Asia Prediction pillar projects 3 trajectories, optimistic, median, and adverse, with deterministic probabilities and estimated tipping dates (EQOS Asia method page, as of October 2026). The frame covers the whole situation and not only the payoffs of one option.
A reader can try a first score in the simulator on the same decision and place it next to the expected value. When the average looks attractive and the score signals fragility, the pair of results tells the team where to dig before committing.
See also in this guide
Frequently asked questions
What is the expected value formula?
The expected value formula is EV = p1 × x1 + p2 × x2 + … + pn × xn. Each payoff x is multiplied by its probability p, and the products are added. The probabilities of the outcomes of one option sum to 1. The result is the probability weighted average payoff.
How many outcomes should an expected value calculation include?
Three to five outcomes per option are enough for most business decisions. They capture the strong, normal, and weak cases without crowding the sheet. Make the outcomes complete and non-overlapping, and check that their probabilities sum to 1 before multiplying.
Is the highest expected value always the best choice?
Not always. Expected value ignores spread, so an option with the highest average can carry a loss that the organisation cannot absorb. Read the worst case and the standard deviation beside the average, and test the key probability before choosing.
Can the EQOS Asia pre-diagnostic check my probabilities?
The free 30-minute pre-diagnostic applies the EQOS Asia equation to your decision across 9 dimensions and shows which carry the most risk. It runs by video in Singapore time (SGT) under NDA, and the result gives a second view next to your own estimates (EQOS Asia contact page, as of October 2026).
Test your knowledge
1. What must the probabilities of one option sum to?
- 0.5
- 1
- 10
Show the answer
1.
2. What is the expected value of the broad campaign in the example?
- 225
- 200
- 180
Show the answer
225.
3. What is the standard deviation of the partnership?
- 249
- 0
- 45
Show the answer
0.
4. What happens when the strong case falls to 15 percent?
- The campaign still leads
- The partnership leads
- The pilot leads
Show the answer
The pilot leads.
5. What does discounting do to payoffs in later periods?
- Brings them to today’s value
- Raises their value
- Removes them
Show the answer
Brings them to today’s value.
About the author
Stéphane Jambu
President, EQOS Forecast Asia
Stéphane Jambu leads EQOS Forecast Asia and works as a consultant in e-commerce, search visibility, and AI. LinkedIn profile
Sources: EQOS Asia method page (eqos-asia.com/method.html), simulator page (eqos-asia.com/simulator.html), and contact page (eqos-asia.com/contact.html), consulted on 22 October 2026. Page updated on 22 October 2026.