OneTaskBy Tim King · Updated
Expected Value Calculator
Expected Return and Variance for Bets, Games and Decisions, With Working
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How the Expected Value Calculator works
This expected value calculator finds the long-run average of anything with uncertain outcomes. Enter each possible outcome and its probability and it returns the expected value, the variance and the standard deviation, with the full sum written out so you can check it.
The expected value is what you would average per go if you repeated a bet, game or decision many times. A single play can land anywhere, but over hundreds of plays the average settles on the expected value. A negative figure means you lose on average, which is how casinos stay in business.
How the expected value calculator works
The formula is E(X) = Σ x × P(x): multiply each outcome by its probability and add them up. The variance is Σ x² × P(x) − E(X)², and its square root is the standard deviation, a measure of how far single results usually stray. Probabilities must total 1, or 100 if you entered percentages.
First example. The default game costs 5 to lose 60% of the time, breaks even 25% of the time and pays 10 or 50 otherwise. E(X) = −3 + 0 + 1 + 2.5 = 0.5 per play.
Second example. A £1 bet on red in European roulette has E(X) = 18/37 − 19/37 ≈ −2.7p.
How many plays before the average settles? Roughly enough that the SD divided by √plays is small next to the expected value, often thousands.
Expected value of everyday bets, games and decisions
| Situation | Outcomes | Probabilities | Expected value |
|---|---|---|---|
| Fair coin, win 1 or lose 1 | 1, −1 | 0.5, 0.5 | 0 |
| Fair six-sided die | 1–6 | 1/6 each | 3.5 |
| Two dice, total | 2–12 | 1/36 to 6/36 | 7 |
| Roulette, £1 on red (European) | +1, −1 | 18/37, 19/37 | −0.027 |
| Roulette, £1 on red (American) | +1, −1 | 18/38, 20/38 | −0.0526 |
| Roulette, £1 on one number (European) | +35, −1 | 1/37, 36/37 | −0.027 |
| Coin, win 2 or lose 1 | 2, −1 | 0.5, 0.5 | 0.5 |
| Card draw, win 10 on an ace | 10, −1 | 4/52, 48/52 | −0.154 |
| £2 scratchcard, 1 in 5 wins £5 | 3, −2 | 0.2, 0.8 | −1 |
| Raffle, 500 tickets at £1, prize £200 | 199, −1 | 1/500, 499/500 | −0.6 |
| Insurance claim, 1% chance of £10,000 | 10,000, 0 | 0.01, 0.99 | 100 |
| Delivery: on time, 1 day late, 3 days late | 0, 1, 3 | 0.8, 0.15, 0.05 | 0.3 |
| Children per family (sample) | 0, 1, 2, 3, 4 | 0.2, 0.3, 0.3, 0.15, 0.05 | 1.45 |
| Penalty kick, scores 75% | 1, 0 | 0.75, 0.25 | 0.75 |
| Exam multiple choice guess, 4 options, −0.25 wrong | 1, −0.25 | 0.25, 0.75 | −0.0625 |
| Product launch: big hit, modest, flop | 500, 100, −200 | 0.1, 0.5, 0.4 | 10 |
| Sales call outcome | 1200, 0 | 0.08, 0.92 | 96 |
| Warranty: no fault, small, major repair | 0, 50, 400 | 0.85, 0.12, 0.03 | 18 |
| Bus wait, minutes | 0, 5, 10, 15 | 0.1, 0.4, 0.3, 0.2 | 7.5 |
| Three coin flips, number of heads | 0, 1, 2, 3 | 1/8, 3/8, 3/8, 1/8 | 1.5 |
| Darts, points per throw (casual) | 0, 5, 20, 50 | 0.2, 0.5, 0.25, 0.05 | 6.5 |
| Lottery, £2 ticket, 1 in 14 m jackpot of £5 m | 4,999,998, −2 | 1/14 m, rest | ≈ −1.64 |
Tips
- Fractions work. Enter 1/6 or 0.1667; both parse.
- Include the stake. Put net outcomes, such as −1 for a lost £1 bet, not the gross payout.
- High SD means high risk. Two choices with the same expected value can feel very different.
FAQ
What does a negative expected value mean?
It means you lose money or value on average over many repeats. Almost every casino game, lottery and raffle has a negative expected value for the player. A single play can still win, but the more you play, the closer your average result gets to the expected loss per play.
Can the expected value be an outcome that never happens?
Yes. The expected value of a fair die roll is 3.5, which no roll can produce. It is an average over many repeats, not a prediction of any single result, in the same way that an average family can have 2.4 children even though no real family does.
How is expected value used in business?
It weighs uncertain options on a common scale. Multiply each possible profit or loss by its estimated probability and add them. The option with the highest expected value wins on average, though a business may still prefer a lower-risk option if a bad outcome would be ruinous.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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