Answer first
What this calculator tells you
Calculate the expected value of a bet or decision from its chance of winning, the amount won and the amount lost. Find the average result of a repeated gamble before you take it. Formula: Expected value = (chance of winning × amount won) − (chance of losing × amount lost). At the worked-example inputs, the expected value per attempt is $40. Holding every other input steady, moving chance of winning from 36.0% to 44.0% moves the result from $26 to $54.
Transparent method
The formula
Find the average result of a repeated gamble before you take it.
Worked example
Example inputs
How to interpret the result
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.
At the worked-example inputs the expected value per attempt is $40. It rises with amount won and chance of winning and falls as amount lost increases.
These figures assume the model's conditions hold: independent trials with a fixed chance for the binomial, and a bell-shaped distribution for the normal. Real data that breaks those assumptions will not follow the calculated probability.
Before you rely on it
What to check
Check that the chance and the amounts describe the same event. A win chance from a different game paired with these payoffs gives a number that means nothing.
The common error
Where people go wrong with expected value calculator
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.
Sensitivity evidence
How chance of winning changes the expected value per attempt
Holding every other input at the worked-example value, moving chance of winning from 36.0% to 44.0% moves the expected value per attempt from $26 to $54: a spread of $28, or 70% of the worked-example result.
| Chance of winning | Expected value per attempt | Expected value over 100 attempts |
|---|---|---|
| 36.0% | $26 | $2,600 |
| 38.0% | $33 | $3,300 |
| 40.0%worked example | $40 | $4,000 |
| 42.0% | $47 | $4,700 |
| 44.0% | $54 | $5,400 |
Every input, tested
Which input moves the expected value per attempt most
Of the 3 inputs, amount won moves the expected value per attempt most ($20 across the range tested) and amount lost moves it least ($12).
| Input | Tested from | To | Expected value per attempt at each end | Swing |
|---|---|---|---|---|
| Amount won | $225 | $275 | $30 to $50 | $20 (50%) |
| Chance of winning | 38.0% | 42.0% | $33 to $47 | $14 (35%) |
| Amount lost | $90 | $110 | $46 to $34 | $12 (30%) |
Two variables at once
Expected value per attempt by chance of winning and amount won
Across the grid the expected value per attempt runs from $8.00 to $76. Moving chance of winning from 36.0% to 44.0% shifts it by $28 at the middle column, and moving amount won from $200 to $300 shifts it by $40 at the middle row, so amount won is the bigger lever here.
| 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 |
The highlighted cell is the worked example: $40.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Chance of winning | 40.0% | Enter the chance of winning used in this calculation. |
| Amount won | $250 | Enter the amount won used in this calculation. |
| Amount lost | $100 | Enter the amount lost used in this calculation. |
| Expected value per attempt | $40 | |
| Expected value over 100 attempts | $4,000 | |
Inputs, definitions and assumptions
Chance of winning
Enter the chance of winning used in this calculation. The prefilled worked-example value is 40.0%.
Amount won
Enter the amount won used in this calculation. The prefilled worked-example value is $250.
Amount lost
Enter the amount lost used in this calculation. The prefilled worked-example value is $100.
How to use this calculator
- 1Verify the inputs. Gather chance of winning, amount won and amount lost from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the expected value per attempt at $40. Store your own version of it as Scenario A.
- 3Test one change. Start with amount won, the input with the biggest effect here: moving amount won from $225 to $275 takes the expected value per attempt from $30 to $50, a swing of 50% of the worked-example figure.
- 4Check the boundary. Read the interpretation boundary above before acting on the result.
People also ask
Frequently asked questions
How do you calculate expected value?
Expected value = (chance of winning × amount won) − (chance of losing × amount lost). Enter chance of winning in percent, amount won in dollars and amount lost in dollars (40 means 40%). At the worked-example inputs the expected value per attempt is $40.
What does the expected value result mean?
Find the average result of a repeated gamble before you take it. At the worked-example inputs the expected value per attempt is $40. It rises with amount won and chance of winning and falls as amount lost increases.
How much does chance of winning change the expected value per attempt?
Holding every other input at the worked-example value, moving chance of winning from 36.0% to 44.0% moves the expected value per attempt from $26 to $54, a spread of $28.
What are the limits of this expected value calculator?
These figures assume the model's conditions hold: independent trials with a fixed chance for the binomial, and a bell-shaped distribution for the normal. Real data that breaks those assumptions will not follow the calculated probability. The tables on this page test chance of winning only from 36.0% to 44.0%; a value outside that range is not tabulated here.
Which input moves the expected value per attempt most in the expected value calculator?
Ranked by how far each moves the expected value per attempt across the range tested: amount won ($20, 50%), chance of winning ($14, 35%) and amount lost ($12, 30%).
How much does amount won matter in the expected value calculator?
The worked example uses $250. With the other inputs left at the worked example, moving amount won from $225 to $275 takes the expected value per attempt from $30 to $50, a swing of 50% of the worked-example figure.
How much does amount lost matter in the expected value calculator?
The worked example uses $100. Holding every other input at its worked-example value, moving amount lost from $90 to $110 takes the expected value per attempt from $46 to $34, a swing of 30% of the worked-example figure.
Which inputs change the expected value over 100 attempts in the expected value calculator?
At the worked-example inputs it is $4,000. Chance of winning takes it from $3,300 to $4,700, amount won takes it from $3,000 to $5,000 and amount lost takes it from $4,600 to $3,400.
Why use median instead of mean?
Mean is pulled toward extreme values. Median is not. A neighborhood where one mansion sits among modest homes has a mean price far above what a typical home costs, while the median reflects the typical home much better. Large gaps between mean and median signal a skewed distribution.
What's the difference between population and sample standard deviation?
Population standard deviation divides by n and is used when your data IS the entire group you care about. Sample standard deviation divides by (n−1) and is used when your data is a subset meant to estimate a larger population: the smaller divisor corrects for the fact that a sample tends to understate true variability.
Sources and evidence
Free Calculators Online is independent and is not affiliated with or endorsed by the source organizations. Educational estimates only.