In short: Expected value is the average result of a decision if you could repeat it many times. You multiply each possible outcome by its probability, then add the results. It helps you compare uncertain choices clearly, though it has limits: a positive average does not make a risk acceptable if one outcome could ruin you.
The basic calculation
Expected value, also called expectation or the mean, gives the long-term average of an uncertain outcome. The method has two steps: weight each possible result by its probability, then add the results together.
Consider a side project that could go one of three ways. All these numbers are illustrative.
| Outcome | Probability | Result | Probability × result |
|---|---|---|---|
| Does well | 20% | +$30,000 | +$6,000 |
| Modest | 50% | +$4,000 | +$2,000 |
| Fails | 30% | -$5,000 | -$1,500 |
| Expected value | +$6,500 |
You will never actually receive $6,500. You will get one of the three results. Its use is as a way to summarize a messy set of possibilities, so you can compare this project with other uses of your time and money.
Four places it helps
Insurance and warranties. A $150 extended warranty on a device with an estimated 10% chance of a $600 repair has an expected benefit of $60. You pay $150 for $60 of expected value. Products like this are usually priced in the seller’s favor, so they make sense mainly when you could not absorb the loss. That applies much more to a house fire than to a blender, and it also explains why insuring catastrophic risk can be sensible even though the expected value is negative. See what a deductible is for a related decision.
Paying down debt or investing. Paying off a card charging 21% yields a certain 21% saving, so its expected return is close to that figure. A stock investment has a lower expected return and a wide range of results. Expected value makes this comparison concrete, which is why debt repayment usually comes first.
Job offers. A startup offering $20,000 less in salary plus equity is a bet you can estimate. Suppose there is a 10% chance the equity is worth $400,000 and a 90% chance it is worth nothing. Its expected value is $40,000 in total, spread over the years it takes to vest. Set against a four-year salary gap of $80,000, the offer looks worse on money alone. Your own probabilities may differ, and that is precisely what the exercise forces you to examine.
Everyday time decisions. Is a two-hour wait worth a 25% chance of getting a $200 refund? The expected value is $50, or $25 an hour. Put that way, the question is easier to answer.
When the average misleads
Expected value is useful but incomplete, and treating it as the final word can do real damage.
Ruin matters more than averages. A bet offering a 95% chance of doubling your net worth and a 5% chance of losing all of it has a strongly positive expected value. You still should not make it if you cannot recover, because you only live through one outcome. A positive average does nothing for someone who has been wiped out.
One shot differs from many. The average describes what happens over many repetitions. For a once-in-a-lifetime decision, the average may never appear in your own life. It is safer to look at the full range, particularly the worst case.
Dollars do not all feel equal. Your first $50,000 matters much more than your tenth. Losing a large sum generally hurts more than winning the same sum helps, a tendency called loss aversion. That reaction can be excessive in small stakes, yet it is often reasonable when the amount is large relative to your wealth.
The probabilities are often guesses. Precise figures can create a false sense of certainty. Garbage in, garbage out applies here too. When you are unsure, test the calculation with more pessimistic numbers and see whether the conclusion survives.
Howard Marks, the investor, recommends asking about the range of likely outcomes and the probability that you are right. His questions, which we expand on in second-order thinking, are a useful companion to this arithmetic.
A practical routine
For a meaningful financial decision, use this five-step routine.
- List the realistic outcomes, including the bad ones you would rather ignore.
- Assign a rough probability to each, and make sure they add up to 100%.
- Calculate the expected value for each option.
- Check the worst case. Could you live with it?
- Run it again with pessimistic numbers.
If an option has a good expected value and an acceptable worst case, you have a strong candidate. If the expected value is good but the worst case is ruinous, shrink the bet or insure against the bad outcome. Combined with awareness of anchoring bias, this routine reduces the influence of the first number you hear. It forms part of the decision-making habits in our guide to personal finance basics and the psychology of money.
Frequently asked questions
What is the formula for expected value?
It is the sum, across all possible outcomes, of each outcome’s value multiplied by its probability. Written out, that is EV = (P1 × X1) + (P2 × X2) and so on, where the probabilities add up to 100%. The calculation is simple, and the hard part lies in estimating the probabilities honestly.
Why do people buy lottery tickets if the expected value is negative?
For most tickets, the expected value is well below the price, so buyers lose money on average. They may be paying for something else, such as a small hope or some entertainment. A modest amount for fun is a fair choice, but it should not be mistaken for an investment.
Should I make every decision using expected value?
No. It suits decisions with money at stake and outcomes you can roughly estimate. Personal matters, such as who to marry or where to live, involve values and consequences that resist this kind of calculation. Even so, listing the outcomes and being honest about their likelihood can clarify your thinking.
How is expected value different from probability?
Probability tells you how likely an outcome is, and expected value also accounts for how large the outcome would be. An unlikely event can dominate the decision if its size is large enough, and a likely gain may matter little if it is very small. Considering both together usually changes which option looks best.
