Expected Value: formula
Expected value ignores the spread of outcomes, so a positive-value bet can still ruin you if a loss is large relative to your bankroll. Probabilities in real life are estimates, and errors change the answer. People value gains and losses unequally, so many decline positive-value bets and accept negative ones.
The average applies over many trials, not one.
Expected Value: worked example
| Chance of winning | 40.0% |
|---|---|
| Amount won | $250 |
| Amount lost | $100 |
| Expected value per attempt | $40 |
| Expected value over 100 attempts | $4,000 |
Enter the chance of winning as a percentage, the amount you win and the amount you lose. The chance of losing is the remainder. Include all costs in the loss, such as the price of a ticket or a fee, and only count amounts that change hands.
For decisions with more than two outcomes, compute each outcome's probability times payoff and add them up.
Expected Value: changing Chance of winning
| Chance of winning | Expected value per attempt | Expected value over 100 attempts |
|---|---|---|
| 36.0% | $26 | $2,600 |
| 38.0% | $33 | $3,300 |
| 40.0% (example) | $40 | $4,000 |
| 42.0% | $47 | $4,700 |
| 44.0% | $54 | $5,400 |
Expected Value: compare input ranges
| Input | Tested from | To | Expected value per attempt: low input | Expected value per attempt: high input | Swing |
|---|---|---|---|---|---|
| Amount won | $225 | $275 | $30 | $50 | $20 (50%) |
| Chance of winning | 38.0% | 42.0% | $33 | $47 | $14 (35%) |
| Amount lost | $90 | $110 | $46 | $34 | $12 (30%) |
Expected Value: individual input comparisons
| Input | Value | Expected value per attempt | Expected value over 100 attempts |
|---|---|---|---|
| Chance of winning | 36.0% | $26 | $2,600 |
| 38.0% | $33 | $3,300 | |
| 40.0% | $40 | $4,000 | |
| 42.0% | $47 | $4,700 | |
| 44.0% | $54 | $5,400 | |
| Amount won | $200 | $20 | $2,000 |
| $225 | $30 | $3,000 | |
| $250 | $40 | $4,000 | |
| $275 | $50 | $5,000 | |
| $300 | $60 | $6,000 | |
| Amount lost | $80 | $52 | $5,200 |
| $90 | $46 | $4,600 | |
| $100 | $40 | $4,000 | |
| $110 | $34 | $3,400 | |
| $120 | $28 | $2,800 |
Expected Value: combined input changes
| Chance of winning / Amount won | $200 | $250 | $300 |
|---|---|---|---|
| 36.0% | $8.00 | $26 | $44 |
| 38.0% | $14 | $33 | $52 |
| 40.0% | $20 | $40 | $60 |
| 42.0% | $26 | $47 | $68 |
| 44.0% | $32 | $54 | $76 |
Expected Value: which assumptions move the result?
Expected Value: values and definitions
| Quantity and definition | Value used |
|---|---|
| Chance of winning | 40.0% |
| Amount won | $250 |
| Amount lost | $100 |
| Expected value per attempt | $40 |
| Expected value over 100 attempts | $4,000 |
Common error: Reading a positive expected value as a sure gain. A single attempt can still lose the full amount, and a run of losses can come first.
Statistics & Probability calculators: Bayes' Theorem Calculator, Mean & Median Calculator, Standard Deviation Calculator, Range & Midrange Calculator, Permutation & Combination Calculator, Simple Probability Calculator.
Expected value is the average result if you took the same bet many times. A 40 percent chance to win $250 against a 60 percent chance to lose $100 averages +$40 each time, or +$4,000 over a hundred tries. A positive expected value does not mean any one try wins.
It means the average wins, if the odds are what you think they are.
Expected value combines the probability and payoff of the same defined event. A win chance from a different game paired with these payoffs gives a number that means nothing.
See sizing bets on the Kelly criterion calculator, chances on the probability and binomial pages, and risk-reward for trades on the risk reward ratio calculator. The Bayes theorem calculator updates probabilities. The option profit calculator uses payoff ideas for options.
What is expected value?
The average outcome of a random event over many repetitions, found by multiplying each outcome by its probability and adding them up.
Does a positive expected value guarantee a profit?
No. It gives the long-run average, and a single bet or a short streak can lose money.
Why do lotteries have negative expected value?
Because the prize pool returns only part of the ticket sales, so the average ticket is worth less than its price.