# Expected Value: How to Make Better Bets in Life, Work and Money

> Expected value weighs each outcome by its probability to compare choices. Learn the formula, worked examples, and when maximizing expected value goes wrong.

Canonical: https://mentarium.io/mental-models/expected-value.md

Mental model

6 min read By [Nemo](https://mentarium.io/about) Updated October 30, 2026

> **Expected value** is the average result you would get from a choice if you could make it many times: each possible outcome multiplied by its probability, then added up. It lets you compare risky options on one number, as long as you also check the worst case, since a good average can hide a ruinous outcome.

- **Category**: Mental model (probability and decision-making)
- **Origin**: Pascal and Fermat, 1654; Christiaan Huygens, 1657
- **Best used for**: Comparing risky options, pricing bets, choosing projects
- **Related**: Probabilistic thinking, base rates, margin of safety

## Ask yourself

- What are the possible outcomes, and roughly how likely is each?
- If I made this choice 100 times, what would the average result be?
- Is there any outcome I could not survive, whatever the average says?
- Am I rejecting a good bet just because I might lose this one time?
- Which probability or payoff in my estimate am I least sure about?

## What is expected value?

Expected value is the average outcome of a choice if you could repeat it many times. You list the possible outcomes, multiply each by its probability, and add them up.

**Expected value = Σ (probability × payoff)**

Say a project has a 60% chance of making $10,000 and a 40% chance of losing $5,000. Its expected value is 0.6 × 10,000 + 0.4 × (−5,000) = 6,000 − 2,000 = $4,000. You will never actually get $4,000 from one try: you get either $10,000 or −$5,000. But if you could run this project many times, you would average about $4,000 a run.

The model’s value is that it puts uncertain options on one scale. A safe $3,000 and a risky project with an expected value of $4,000 can now be compared directly, and then you can ask whether the extra $1,000 is worth the risk.

## Where it comes from

Expected value was born from gambling. In 1654, Blaise Pascal and Pierre de Fermat exchanged letters about the “problem of points”: if a game of chance is interrupted, how should the stakes be split fairly? Their answer was to weigh each possible way the game could have finished by its probability. In 1657, Christiaan Huygens published a short treatise on games of chance that set out the concept formally.

The first famous problem with it came soon after. In the **St. Petersburg paradox**, posed by Nicolaus Bernoulli in 1713, a coin is flipped until it lands heads, and the payout doubles with each flip. The expected value is infinite, yet almost nobody would pay more than a modest sum to play. In 1738, his cousin Daniel Bernoulli proposed that people care about the usefulness of money, not the amount, and that an extra dollar matters less the more you have. That became expected utility, the basis of much of modern economics.

## How to use expected value

1. **List the outcomes.** Write down the realistic results of each option. Three to five is usually enough: great, okay, bad, and anything catastrophic.
2. **Estimate the probabilities.** Start from [base rates](https://mentarium.io/mental-models/base-rates) where you can: how often do similar choices land in each outcome? Make sure the probabilities sum to 100%.
3. **Put a value on each outcome.** Money is easiest, but you can use time, a 1 to 10 satisfaction score, or anything consistent across options.
4. **Multiply and add.** Compute the expected value of each option, including “do nothing,” which usually has a value too (often a cost, through [opportunity cost](https://mentarium.io/mental-models/opportunity-cost)).
5. **Check the tails before you choose.** Look at the worst outcome of the winning option. If you could not survive or recover from it, the average does not matter. Then run a sensitivity check: if you are 10 points wrong on the probability you are least sure of, does the winner change?

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## 3 real-world examples

### Business

Say a small company must choose between two marketing tests, each costing $5,000. Test A has a 50% chance of returning $12,000. Test B has a 10% chance of returning $80,000. Ignoring other outcomes, A’s expected return is $6,000 and B’s is $8,000, so B looks better on average. But B fails nine times out of ten. If the company can run many such tests over a year, B is the better strategy. If this is its only shot, the higher chance of A may matter more. Expected value tells you what to do in the long run; your situation tells you whether you get a long run. (A hypothetical example.)

### Personal life

Applying for a reach job feels risky because you will probably be rejected. Expected value reframes it. If an application takes three hours, there is maybe a 10% chance of an offer, and the job is worth much more to you than your current one over several years, the expected value of applying is clearly positive. Fear of rejection makes the 90% loss feel large; the math shows the loss is three hours. [Loss aversion](https://mentarium.io/biases/loss-aversion) makes people skip bets like this all the time.

### Investing

Investors think in expected value whether they say so or not: an investment is attractive when the probability-weighted outcomes beat the price. Expected value also explains why ruin matters so much. If a strategy has a positive expected value but a small chance of losing everything, repeating it enough times makes that loss likely. This is why a [margin of safety](https://mentarium.io/mental-models/margin-of-safety) and position sizing matter as much as the average return.

## Common mistakes

- **Precise math on made-up numbers.** Expected value is only as good as the probabilities. If you guessed them, treat the result as a rough guide and run the sensitivity check.
- **Ignoring ruin.** A positive average does not help if one outcome ends the game. Survive first, then maximize.
- **Forgetting the “do nothing” option.** Inaction has an expected value too, and it is rarely zero.
- **Comparing one-time and repeated bets the same way.** The law of averages works over many tries. For a once-in-a-life choice, the spread of outcomes matters more.
- **Leaving out non-money values.** Time, stress and reputation are real payoffs. Put a rough number on them rather than pretending they are zero.

## Expected value vs expected utility

|  | Expected value | Expected utility |
|---|---|---|
| What it averages | Raw payoffs (dollars, hours) | How much each payoff is worth to you |
| Origin | Pascal, Fermat, Huygens, 1650s | Daniel Bernoulli, 1738 |
| Treats a gain and an equal loss as | Equal and opposite | Often unequal (losses hurt more when you have less) |
| Best for | Repeated, small, survivable bets | Large or one-time decisions |
| Weak spot | Can recommend ruinous bets | Hard to measure “utility” in practice |
| St. Petersburg game | Says pay any price | Says pay a modest amount |

## The Mentarium take

Expected value is the single most useful number in decision-making, and it is never enough on its own. We use it as a two-step rule: first, kill any option with an outcome you cannot survive; second, among what is left, choose the highest expected value. Most bad financial decisions break step one. Most timid career decisions break step two.

In the [Mentarium Decision Loop](https://mentarium.io/decision-making/decision-making-process), expected value lives in the Map step, where you price each option, and the Decide step, where you write your probability estimates down. That record is what makes the Review step useful: after a dozen decisions you will see whether your 70% calls happen about 70% of the time. If you want a guide through the numbers, the [Decision Coach](https://mentarium.io/coach) runs the Loop with you, and [probabilistic thinking](https://mentarium.io/mental-models/probabilistic-thinking) is the companion model for getting the probabilities right in the first place.

One last practical note: you do not need precise numbers to get most of the benefit. Rough brackets (likely, possible, unlikely; small, medium, large) already beat the usual alternative, which is choosing by whichever outcome feels most vivid. The discipline is in listing all the outcomes and assigning something to each, especially the dull middle ones that people skip. Once the outcomes are written down, the right choice is often obvious before you multiply anything. When it is not, the multiplication is a few minutes of work, and those minutes are cheap compared with a bet made on gut feel alone.

**Counters** [Loss Aversion](https://mentarium.io/biases/loss-aversion)

## FAQ

### What is the expected value formula?

Expected value = (probability of outcome 1 × value of outcome 1) + (probability of outcome 2 × value of outcome 2) + and so on for every outcome. The probabilities must add up to 100%.

### What does a positive expected value mean?

It means that, on average over many repetitions, the choice gains more than it loses. A positive expected value bet can still lose on any single try, which is why you also need to check whether you can afford the bad outcomes.

### What is an example of expected value?

A bet that pays $100 with a 30% chance and costs $20 otherwise has an expected value of 0.3 × 100 + 0.7 × (−20) = 30 − 14 = $16. On average you gain $16 per bet, even though you lose 70% of the time.

### Why don't people always maximize expected value?

Because a single outcome can matter more than the average. Losing your house is worse than the math of an equal-sized gain suggests. Daniel Bernoulli's 1738 answer to the St. Petersburg paradox was to weigh outcomes by their usefulness to you, which became expected utility.

### Who invented expected value?

The idea grew out of the 1654 letters between Blaise Pascal and Pierre de Fermat about how to split the stakes of an unfinished game. Christiaan Huygens published the first formal treatment in 1657.

## Related models & biases

[Mental model Probabilistic Thinking Probabilistic thinking is the habit of treating outcomes as likelihoods rather than certainties: estimating how likely each possibility is, updating those estimates as new evidence arrives, and judging decisions by the quality of the reasoning and the odds at the time, not only by how they turned out.](https://mentarium.io/mental-models/probabilistic-thinking) [Mental model Base Rates and Base Rate Neglect A base rate is how often something happens in a relevant population, before you look at any specific case. Base rate neglect (the base rate fallacy) is ignoring that number in favor of vivid case details. Kahneman and Tversky showed it with the taxicab problem, where most people overweight a witness and ignore how few cabs are Blue.](https://mentarium.io/mental-models/base-rates) [Mental model Margin of Safety A margin of safety is the buffer between what you expect to need and what you actually prepare for, so that a plan still works if your estimates are wrong. Investors buy well below estimated value; engineers build stronger than the expected load; anyone can apply it to time, money and health.](https://mentarium.io/mental-models/margin-of-safety) [Mental model Opportunity Cost Opportunity cost is the value of the best alternative you give up when you make a choice. Every use of money, time or attention rules out other uses, so the real cost of anything is not just what you pay, but what that money, time or attention could have done in its next-best use.](https://mentarium.io/mental-models/opportunity-cost) [Cognitive bias Loss Aversion Loss aversion is the tendency to feel a loss more strongly than a gain of the same size, so that losing $100 hurts more than winning $100 pleases. Described in Kahneman and Tversky's prospect theory, it leads people to reject favorable bets, hold on to losing positions and overvalue what they already own.](https://mentarium.io/biases/loss-aversion)

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