Expected Value Calculator
Mean outcome of a bet or forecast from values and probabilities.
What it does
Expected value prices uncertainty with one number: each outcome times its chance, added across all cases. A lottery ticket, a sales forecast, and a game bet all reduce to the same weighted average, which is the long run mean per repeat.
The tool above handles two or three outcomes. It returns the mean with variance and standard deviation beside it. The default 0, 5, and 10 at chances 0.2, 0.5, and 0.3 gives 5.50 with variance 12.25 and SD 3.50.
Statistics texts state the rule as mu = sum of x times P(x), with worked checks that match this page. The soccer scoring model at 0, 1, 2 with chances 0.2, 0.5, 0.3 gives 1.10, and Penn State uniform 0 to 4 gives 2 with variance 2.
Averages need repeats to bite. One trial can land anywhere in the table while hundreds settle near the mean, and the standard deviation says how wild the ride stays even at that scale.
How to use
- Pick two or three outcomes.
- Type each outcome value, gains positive and losses negative.
- Type each chance as a decimal between 0 and 1.
- Force chances to add to exactly 1.
- Read expected value first, spread beside it.
- Price fair bets at the mean; charge above it for risk.
Formula
- pi
- Chance of outcome i as a decimal; all pi add to 1.
- xi
- Value of outcome i in the same units.
- Var, SD
- Spread of outcomes around the mean.
Two outcome mode drops the third pair entirely. Percent chances convert first: 30% is 0.3.
Worked example
Example: 0, 5, 10 at 0.2, 0.5, 0.3
Defaults: three outcomes as listed.
- Parts: 0 x 0.2 = 0, 5 x 0.5 = 2.5, 10 x 0.3 = 3.
- Mean: 0 + 2.5 + 3 = 5.50.
- Gaps: minus 5.5, minus 0.5, plus 4.5.
- Variance: 30.25 x 0.2 + 0.25 x 0.5 + 20.25 x 0.3 = 12.25.
- SD: root of 12.25 = 3.50.
- Tool shows 5.50, 12.25, 3.50.
Reference table
| Case | Mean | Spread |
|---|---|---|
| 0, 5, 10 at 0.2, 0.5, 0.3 | 5.50 | Var 12.25, SD 3.50 |
| Soccer 0, 1, 2 at 0.2, 0.5, 0.3 | 1.10 | Var 0.49, SD 0.70 |
| Uniform 0 to 4, each 0.2 | 2.00 | Var 2.00, SD 1.41 |
| Win 10 at 0.3, lose 2 at 0.7 | 1.60 | Var 30.24, SD 5.50 |
Limits
- Known chances assumed; guessed chances give guessed means.
- One trial proves nothing; means need repeats.
- Stakes beyond money, like ruin risk, sit outside.
- Skewed tails can dominate while the mean looks tame.
Common mistakes
- Adding percents without converting to decimals.
- Letting chances add to 0.9 or 1.1.
- Reading variance units as outcome units.
- Expecting one trial to land on the mean.
- Ignoring spread when means tie.
Try it yourself
Quick checks. Answers sit tucked under each question.
Win 10 at 0.3, lose 2 at 0.7?
3 minus 1.4 is 1.60; variance 30.24, SD 5.50.
Soccer 0, 1, 2 at 0.2, 0.5, 0.3?
Mean 1.10, variance 0.49, SD 0.70.
Chances 0.5 and 0.4?
Rejected: chances must add to 1, not 0.9.
Questions
What is expected value?
The probability weighted mean: each outcome times its chance, added up.
What is a fair price?
The mean itself. Pay less for an edge; charge more to carry risk.
Negative expected value?
A long run loss per play, like most lottery tickets.
Variance versus SD?
Variance is mean squared gap; SD is its root in outcome units.
Must chances add to 1?
Yes. The tool rejects anything else and prints the actual sum.
Does one trial match the mean?
Rarely. Means describe many repeats, not single shots.
Sources
- OpenStax Introductory Statistics: Mean or Expected Value
- Penn State STAT 500: Probability Distributions