This expected value calculator computes the long-run average of any gamble, game or uncertain payoff: enter each outcome with its probability, and get the expected value, variance and standard deviation — the full risk profile, not just the headline number.
The Average That Governs the Long Run
E(X) = Σ xᵢ · P(xᵢ) — every outcome weighted by its probability. Variance follows as E(X²) − E(X)², and the standard deviation puts the spread back into outcome units. EV is the mathematics our Probability Calculator‘s outputs feed into: probabilities of events are inputs, decisions are outputs. A fair game has EV exactly equal to its entry price; casinos, lotteries and insurance all live in the deliberate gap. Part of our Statistics Calculators collection.
Worked Example (Verified)
The widget’s game: win $100 (10%), win $20 (30%), lose $10 (60%):
- E(X) = 100×0.1 + 20×0.3 + (−10)×0.6 = 10 + 6 − 6 = +$10 — a favorable game (they rarely are; check the source)
- E(X²) = 10,000×0.1 + 400×0.3 + 100×0.6 = 1,180 → Variance = 1,180 − 100 = 1,080, SD ≈ $32.86
- The pair (EV +10, SD 32.9) is the honest description: profitable on average, swingier than the average is worth per single play — bankroll math (Kelly criterion) starts exactly here
- Lottery reality check: a $2 ticket with a 1-in-10M shot at $5M, plus small prizes summing to EV $0.40, prices its hope at −$1.60 per ticket — entertainment, budgeted honestly
Where EV Thinking Earns Its Keep
- Insurance: your premium always exceeds the EV of your claims (the loading funds the company); you buy it anyway because variance hurts — utility, not EV, is the decider for one-shot risks.
- Business decisions: launch/don’t-launch trees multiply probabilities down branches into EVs, then sensitivity-test the inputs.
- The St. Petersburg trap: a coin-flip game with infinite EV that nobody sane pays much to play — the classic proof that EV alone isn’t a decision rule when ruin or utility curves enter.
- Sample thinking: EV is a population mean in disguise; the law of large numbers is its enforcement mechanism — with SD ÷ √n shrinking the realized average around it.
Related Tools
- Probability Calculator — where the probabilities come from
- Poisson Distribution Calculator — full distributions for counted events
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you calculate expected value?
Multiply each outcome by its probability and add: E(X) = ΣxᵢP(xᵢ). The example: 100(0.1) + 20(0.3) − 10(0.6) = +10. Probabilities must sum to 1 — this calculator flags it when they don’t.
Can expected value be a value that never occurs?
Usually is: a fair die’s EV is 3.5, a result no roll can produce. EV is a balance point (the distribution’s center of mass), not a prediction of any single outcome.
What does a negative EV mean?
Played repeatedly at the same terms, you lose that amount per play on average. A −$0.50 EV on a $1 lottery ticket means the game returns ~50¢ per dollar over time — individually random, collectively certain.
Why include variance with EV?
Two games can share EV = $0 — one paying ±$1 coin flips, one paying +$1,000,000 or −$1,000,000 on a hair’s probability. Variance (here via E(X²) − EV²) and SD are the numbers that tell them apart; risk lives there.
How many trials before EV “kicks in”?
There’s no threshold — the average’s standard error falls as SD/√n: 100 plays of the worked example typically average within ±$3.3 of the $10 EV; 10,000 plays tighten that to ±$0.33. Convergence is gradual, guaranteed only in the limit.
Is this expected value calculator free?
Yes — free, instant, and your numbers stay in the browser.